Entwined Hom-modules were introduced by Karacuha in [13], which can be viewed as a generalization of Doi-Hom Hopf modules and entwined modules. In this paper, the sufficient and necessary conditions for the forgetful functor F : H (M k)(ψ) C A → H (M k) A and its adjoint G : H (M k) A → H (M k)(ψ) C A form a Frobenius pair are obtained, one is that A ⊗ C and the C * ⊗ A are isomorphic as (A; C * op #A)-bimodules, where (A, C, ψ) is a Hom-entwining structure. Then we can describe the isomorphism by using a generalized type of integral. As an application, a Maschke type theorem for entwined Hom-modules is given.