In this work we study, from the numerical point of view, a problem involving one-dimensional thermoelastic mixtures with two different temperatures; that is, when each component of the mixture has its own temperature. The mechanical problem consists of two hyperbolic equations coupled with two parabolic equations. The variational problem is derived in terms of product variables. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced using the finite element method and the backward Euler scheme. A discrete stability property is proved and a priori error estimates are obtained, from which the linear convergence is deduced. Finally, some numerical simulations are described to show the accuracy of the approximation and the behavior of the solution. (C) 2017 Elsevier Ltd. All rights reserved.