This is an appeal from the district court’s judgment awarding Timothy Gag-non backpay, liquidated damages, and attorney’s fees and costs under the Fair Labor Standards Act (FLSA), 29 U.S.C. §§ 201-19, against United Technisource, Inc. (UTI) and AIS Tech Services, Inc. (AIS).

I

Gagnon is a skilled craftsman with many years experience in prepping and painting the exterior and interior of aircrafts. When Gagnon began working for UTI, he executed a contract in which UTI agreed to pay Gagnon $5.50 per hour for “straight time” and $20.00 per hour for overtime. ^[1] Although the record indicates differing hourly wage rates for aircraft painters in the area and at the time in which Gagnon was working, ^[2] none are remotely close to the $5.50 per hour that the UTI/AIS contracts ^[3] established as Gagnon’s “straight time” wage. In addition to his straight time wage, UTI also agreed to pay Gagnon $12.50 for every hour he worked each week up to forty hours per week or a maximum of $500.00. The contract referred to this additional hourly pay as “per diem.”

About a year after he began working for UTI/AIS, Gagnon received a memo that notified him of a “raise in all pay.” The memo noted that “[w]e are pleased to announce that our client [Wing Aviation] has authorized a $1.00 per hour raise in all pay starting this pay check.” To effectuate the raise, however, Gagnon was not given an increase in his “straight time” pay rate of $5.50 per hour. Rather, he received a $1.00 raise in his hourly per diem for all hours worked under forty each week and a $1.00 increase in his overtime rate. The record does not indicate that this increase in hourly per diem was based on any reasonably approximated increase in Gagnon’s expenses.

Eventually, Gagnon filed suit against[*1040] UTI/AIS and Wing Aviation, ^[4] claiming violations of the Americans with Disabilities Act (ADA), the Family and Medical Leave Act (FMLA), and the FLSA. All three Defendants answered and UTI/AIS filed a counterclaim alleging breach of contract and fraud against Gagnon. UTI/AIS contended that Gagnon breached the employment contract and committed fraud by failing to notify UTI/AIS when he moved from a city 280 miles from the Wing Aviation facility to one only nine miles away, and by continuing to receive per diem pay after the move.

Defendants moved for summary judgment on all claims. The district court granted summary judgment to Defendants on the ADA and FMLA claims, ^[5] and found in Gagnon’s favor on the FLSA overtime claim. The district court held that “the per diem allowance is to be included in Gagnon’s regular rate of pay,” and ordered Defendants to “recalculate Gagnon’s rate of pay, determine the credit that is due based on his relocation, and submit same to Gagnon and the Court for review.” The district court did not address UTI/AIS’s counterclaim.

In response to the court’s order, Defendants filed a calculation that excluded “the per diem allowance improperly received ... during and after October 2005” when Gagnon moved closer to Wing Aviation’s facility, and that included a request that the district court order Gagnon to pay $8,150.49 for “the per diem improperly received ... during and after October 2005.” Gagnon filed numerous responsive pleadings as well as his own motion for entry of judgment and for liquidated damages that sought back overtime pay of $4,266.82. Gagnon also moved for attorney’s fees and costs.

The district court denied UTI/AIS’s motion and request, finding “them to be contrary to the Court Memorandum Opinion and rulings from the bench.” After considering the motions and all competent evidence offered by the parties, the district court entered judgment awarding Gagnon back pay of $4,266.82 and, finding UTI/ AIS’s violations willful, liquidated damages of $4,266.82. Over UTI/AIS’s objections, the district court also awarded Gagnon $55,908 in attorney’s fees and $3,568.57 in costs. This appeal followed.

II

We review the district court’s grant of summary judgment de novo. Ackermami v. Wyeth Pharmaceuticals, 526 F.3d 203, 207 (5th Cir.2008). Summary judgment is proper when there is an absence of genuine issues of material fact and one party is entitled to judgment as a matter of law. Thurman v. Sears, Roebuck & Co., 952 F.2d 128, 131 (5th Cir.1992). Thus, “[t]he appropriate inquiry is ‘whether the evidence presents a sufficient disagreement to require submission to a [factfinder] or whether it is so one-sided that one party must prevail as a matter of law.’ ” Septimus v. University of Houston, 399 F.3d 601, 609 (5th Cir.2005) (quoting Anderson v. Liberty Lobby, Inc., 477 U.S. 242, 251-52, 106 S.Ct. 2505, 91 L.Ed.2d 202 (1986)).

A

UTI/AIS argue that their payment scheme does not violate the FLSA because the FLSA only requires employers to pay[*1041] overtime at a rate of time and a half, and UTI/AIS paid Gagnon overtime at a rate more than three times his base pay. UTI/ AIS also argue that Gagnon’s per diem reasonably approximated his reimbursable expenses and should therefore be excluded from the determination of Gagnon’s regular rate for the purposes of overtime pay. According to UTI/AIS, “[i]t cannot be argued ... [that] the per diem was a ploy to avoid paying Gagnon overtime compensation.” We disagree.

The FLSA requires that non-exempt employees who work more than forty hours in a work week must be paid one and one-half times their “regular rate” of pay. 29 U.S.C. § 207(a)(1). The FLSA broadly defines “regular rate” as the hourly rate actually paid the employee for “all remuneration for employment.” 29 U.S.C. § 207(e); see also Walling v. Helmerich & Payne, Inc., 323 U.S. 37, 42, 65 S.Ct. 11, 89 L.Ed. 29 (1944). “The regular rate by its very nature must reflect all payments which the parties have agreed shall be received regularly during the workweek, exclusive of overtime payments.” Bay Ridge Operating Co. v. Aaron, 334 U.S. 446, 461, 68 S.Ct. 1186, 92 L.Ed. 1502 (1948). The “regular rate” becomes a mathematical computation once the parties have decided on the amount of wages and the mode of payment, which is unaffected by any designation to the contrary in the wage contract. Id. The “regular rate” is not an arbitrary label — it is an actual fact. Id.

Here, UTI/AIS have tried to avoid paying Gagnon a higher “regular rate” by artificially designating a portion of Gag-non’s wages as “straight time” and a portion as “per diem.” Although per diem can be excluded from an employee’s regular rate, 29 U.S.C. § 207(e)(2); see also 29 C.F.R. § 778.217(b), the “ ‘regular rate’ of pay ... cannot be left to a declaration by the parties as to what is to be treated as the regular rate for an employee; it must be drawn from what happens under the employment contract.” 29 C.F.R. § 778.108 (citing Bay Ridge Operating Co., 334 U.S. at 465, 68 S.Ct. 1186). The Department of Labor has recognized that when, as here, the amount of per diem varies with the amount of hours worked, the per diem payments are part of the regular rate in their entirety. ^[6]

Furthermore, we are suspicious of UTI/ AIS’s claims that Gagnon’s employment contracts were not a scheme to avoid paying overtime. It is difficult to believe that a skilled craftsman would accept a wage so close to the minimum wage when the prevailing wage for similarly skilled craftsmen was approximately three times the minimum wage. We are similarly troubled by the fact that the combined “straight time” and “per diem” hourly rates approximately match the prevailing wage for aircraft painters. Further, it is suspect that a “raise in all pay” was effectuated by increasing the hourly “per diem” rate rather than the “straight time” rate. Finally, we can conceive of no reason why a legitimate per diem would vary by the hour and be capped at the forty-hour mark, ^[7] which not-[*1042] so-coineidentally corresponds to the point at which regular wages stop and the overtime rate applies.

We find this case analogous to other cases in which employers have sought to artificially lower an employee’s regular rate by mischaracterizing a portion of it as a bonus or where employees were paid low “straight rates” for the first hour or two worked — usually set around minimum wage — after which they earned one and one half times the straight rate, and were consequently paid no premium for their actual overtime work. See Walling v. Youngerman-Reynolds Hardwood Co., 325 U.S. 419, 425, 65 S.Ct. 1242, 89 L.Ed. 1705 (1945); see also 29 C.F.R. § 778.502.

We hold that Gagnon’s hourly per diem allowances of $12.50 and $13.50 were part of his hourly “remuneration for employment” and must be considered in his regular rate for the purpose of determining overtime pay due under the FLSA. Helmerich & Payne, 323 U.S. at 42, 65 S.Ct. 11. Accordingly, we affirm the district court’s determination that UTI/AIS violated the FLSA by not including Gag-non’s per diem in their calculation of his regular rate.

B

Having concluded that the hourly per diem is part of Gagnon’s base pay, we turn to UTI/AIS’s contract and fraud counterclaims. UTI/AIS argue that Gag-non breached the employment agreements by not reporting that he moved closer to the work site and by continuing to receive per diem when he lived less than ten miles from the work site. The district court did not address UTI/AIS’s counterclaims. Although our conclusion that the hourly per diem wages must be included in base pay would seem to eviscerate these claims, our precedent suggests that such claims should not be addressed in a FLSA action. See Brennan v. Heard, 491 F.2d 1, 4 (5th Cir.1974), rev’d on other grounds, McLaughlin v. Richland Shoe Co., 486 U.S. 128, 108 S.Ct. 1677, 100 L.Ed.2d 115 (1988) (noting that the only function of the federal judiciary under the FLSA “is to assure to the employees of a covered company a minimum level of wages” and holding that “[arguments and disputations over claims against those wages are foreign to the genesis, history, interpretation, and philosophy of the Act.”). Accordingly, we find no error in the district court’s decision not to address UTI/AIS’s counterclaims.

C

UTI/AIS argue that the district court erred in finding a willful violation of the FLSA. UTI/AIS argue that “the employment agreements between Gagnon and UTI and AIS were not a deliberate scheme to evade FLSA’s overtime requirements,” and that because Gagnon did not give UTI/ AIS enough information about his address change or how much he required in per diem, “neither UTI nor AIS had the opportunity to comply with the FLSA, and[*1043] could not therefore have -willfully violated the statute with respect to Gagnon.”

The district court’s conclusion that UTI/ AIS acted willfully “is a finding of fact not to be set aside unless found to be clearly erroneous.” Reich v. Tiller Helicopter Sewices Inc., 8 F.3d 1018, 1036 (5th Cir. 1993). “A finding is ‘clearly erroneous’ when although there is evidence to support it, the reviewing court on the entire evidence is left with the definite and firm conviction that a mistake has been committed.” United States v. U.S. Gypsum Co., 333 U.S. 364, 395, 68 S.Ct. 525, 92 L.Ed. 746 (1948). We see no clear error in the district court’s finding that the violation was willful.

D

UTI/AIS also seek an offset in an amount equal to the damages which they incurred because Gagnon breached his contractual obligation to notify UTI/AIS of his address change. They cite Singer v. City of Waco, 324 F.3d 813, 828 n. 9 (5th Cir.2003), for the proposition that “an offset is permissible in an FLSA case unless the result would be a sub-minimum wage.”

Initially, we note our hesitancy to address this claim as it essentially reiterates the same contract counterclaim that we found inappropriate in a FLSA action. Despite this hesitancy, we nonetheless address the claim because we have previously held that offsets are permissible in FLSA actions, see id., and it is plausible that the offset claim seeks to prevent unjust enrichment rather than enforce the terms of the contract.

While Singer does allow offsets, it is distinguishable from this case, as it dealt with an offset that occurred because the city had already paid a large portion of the back overtime pay due to the workers. Id. at 828. The Singer court specifically stated that the offset “simply acknowledged that the City already paid the bulk of its overtime obligations.” Id. In this case, we have concluded that Gagnon’s per diem was part of his “regular rate.” Since the money that the employment contracts labeled as per diem was actually part of Gagnon’s regular rate rather than reimbursement for his work-related expenses, it did not depend on where he lived. The “per diem” that Gagnon received after he moved closer to the Wing Aviation facility was simply regular rate wages to which he was entitled. Accordingly, in paying the per diem, UTI/AIS did not pay Gagnon any additional sums that could be characterized as advanced or inappropriate amounts subject to an offset against the overtime owed to him. This same reasoning also disposes of UTI/AIS’s argument that the per diem that Gagnon received after he moved to within ten miles of the work site should be excluded when calculating his overtime pay.

E

UTI/AIS also argue that the attorney’s fee award was excessive. They argue that both the factual finding as to amount and the decision not to reduce the amount based on the Johnson ^[8] factors were erroneous.

The FLSA provides that reasonable attorney’s fees and costs are penalties that shall be awarded to the employee. 29 U.S.C. § 216(b). We “review the district court’s award of attorney’s fees for abuse of discretion and its factual findings for clear error.” Singer, 324 F.3d at 829.

[*1044] The documentation supporting a factual finding regarding the amount of attorney’s fees must be sufficient for the court to verify that the applicant has met its burden of establishing an entitlement to a specific award. Louisiana Power & Light Co. v. Kellstrom, 50 F.3d 319, 324 (5th Cir.1995). “[C]ourts customarily require the applicant to produce contemporaneous billing records or other sufficient documentation so that the district court can fulfill its duty to examine the application for noncompensable hours,” but “[flailing to provide contemporaneous billing statements does not preclude an award of fees per se, as long as the evidence produced is adequate to determine reasonable hours.” Id. at 325. Although Gagnon has not produced contemporaneous billing records, he has produced very detailed invoices and affidavits that describe the time Gagnon’s attorney spent on the various claims. Having looked through the invoices and affidavits, we see no evidence of the clear error required to disturb the district court’s finding of the lodestar amount. See U.S. Gypsum Co., 333 U.S. at 395, 68 S.Ct. 525 (declining to reverse factual finding absent “the definite and firm conviction that a mistake has been committed”).

Having concluded that the factual determination of amount was not clear error, we turn to whether the district court abused its discretion in not reducing the lodestar calculation based on the Johnson factors. ^[9] In this case, the district court failed to provide any indication that it considered them at all. See Migis v. Pearle Vision, 135 F.3d 1041, 1047 (5th Cir.1998). Arguably, the district court’s award of the exact amount requested by Gagnon could be interpreted as acceptance of the analysis of Johnson factors found in Gagnon’s motion for attorney’s fees. See United States v. Evans, 587 F.3d 667, 673 (5th Cir.2009) (assuming that the district court considered 18 U.S.C. § 3553(a) factors presented to it in briefing even though they were not specifically discussed in ruling). Here, however, where the fee award was more than six times greater than the amount of relief awarded to Gagnon, we cannot simply assume that the district court considered the Johnson factors. See Migis, 135 F.3d at 1047-48; Saizan v. Delta Concrete Products Co., 448 F.3d 795, 800 (5th Cir.2006).

To be clear, this conclusion in no way implies that the attorney’s fee award, if justified by a proper explanation, would be an abuse of discretion. It simply indicates that, without any factual findings, it is impossible to determine whether the district court “sufficiently considered the appropriate criteria.” Saizan, 448 F.3d at 800 (internal quotation marks omitted). Because we cannot properly evaluate the district court’s decision, we must vacate the attorney’s fee award and remand to the district court for consideration of the Johnson factors.

We also note that there remains the question of attorney’s fees for this appeal. “An additional fee to compensate counsel for their services in connection[*1045] with the appeal can be awarded in a [FLSA] case when the appellate court considers such an award appropriate.” Montalvo v. Tower Life Building, 426 F.2d 1135, 1150 (5th Cir.1970) (listing cases). We consider such an award appropriate in this ease and instruct the district court to include these fees in its determination.

F

Finally, UTI/AIS argue that the district court abused its discretion in awarding costs that are not authorized by 28 U.S.C. § 1920. Specifically, UTI/AIS argue that the district court should not have awarded costs for (1) private process servers — $177.50; (2) AT&T teleconference — $32.64; (3) a private investigator— $250.00; (4) Fairway Deliver Services— $6.87; (5) postage — $81.82; (6) FLSA book — $291.99; (7) Lexis-Nexis — $850.77; (8) Secretary of State — $20.50; and (9) Pacer — $4.87, and that the copying costs were not properly justified. “An award of costs is reviewed for abuse of discretion.” Mota v. University of Texas Houston Health Science Center, 261 F.3d 512, 529 (5th Cir.2001). The “Supreme Court has indicated that federal courts may only award those costs articulated in section 1920 absent explicit statutory or contractual authorization to the contrary.” Cook Children’s Medical Center v. The New England PPO Plan of General Consolidation Management Inc., 491 F.3d 266, 274 (5th Cir.2007) (internal quotation marks omitted). The district court did not mention any other statutory or contractual authorization for the costs or explain its decision to award the challenged costs. Thus, we have no basis for determining whether some or all of the challenged costs might fall within the costs-award provision of the FLSA, see 29 U.S.C. § 216(b), or be authorized by some other rationale. See, e.g., Kurtis A. Kemper, Annotation, Recovery of Computer-Assisted Research Costs as Part of or in Addition to Attorney’s Fees Under Federal Fee-Shifting Statutes, 28 A.L.R. FED.2d 397 (2008). Additionally, although § 1920 provides that copying costs can be recovered when copies are obtained for use in the case, the district court made no factual findings as to whether the copies were “necessarily obtained for use in the case.” See Fogleman v. ARAMCO, 920 F.2d 278, 286 (5th Cir.1991). Without a proper basis for evaluating the district court’s award of costs, we cannot determine whether the court abused its discretion. Accordingly, we must vacate the cost award and remand for further proceedings.

Ill

In conclusion, we affirm the district court’s grant of summary judgment in favor of Gagnon. We agree that the per diem pay must be included in Gagnon’s regular rate, that UTI/AIS’s contract and fraud claims cannot be brought as counterclaims in a FLSA action, and that UTI/ AIS is not entitled to any offset for per diem paid after Gagnon moved closer to the Wing Aviation facility. Furthermore, we see no clear error in the district court’s determination that the FLSA violation was willful. We do, however, vacate the award of attorney’s fees and costs. We remand the attorney’s fees and costs determinations to the district court in order for the court to provide additional explanation for its awards and with specific instructions to include time spent on this appeal in determining the amount of fees.

AFFIRMED in part, VACATED in part, and REMANDED.