This paper deals with the diffusive limit of the scaled Goldstein-Taylor model and its approximation by an Asymptotic Preserving Finite Volume scheme. The problem is set in some bounded interval with nonhomogeneous boundary conditions depending on time. We obtain a uniform estimate in the small parameter ε using a relative entropy of the discrete solution with respect to a suitable profile which satisfies the boundary conditions expected to hold as ε goes to 0. Key words: Diffusive limit of hyperbolic systems, initial boundary value problem, finite volume approximation, asymptotic preserving scheme MSC (2010): 65M08, 65M12, 35L50