This talk provides an a priori error analysis of a specific thermoelastic problem governed by the so called Moore–Gibson–Thompson (MGT) equation. This model has been analyzed, over the last ten years, by many researchers and it includes an integral memory term with a regularized kernel but, here, we will assume a finite memory.In this case, the history dependence (memory effects) were incorporated in both the mechanical and thermal components, aligning with Gurtin’s theory to ensure compatibility with the causality principle. Since the existence, uniqueness, and exponential stability of the continuous solution are recalled. From the numerical point of view, the work starts by defining the variational formulation of the problem and the introduction of a fully discrete approximation scheme. This is achieved by using the finite element method for spatial discretization and the implicit Euler scheme for time integration. In the part of the study of the error analysis, it is proved a discrete stability property and derived a priori error estimates, establishing that the proposed algorithm achieves linear convergence under suitable regularity assumptions. Finally, the study concludes with some numerical simulations in one and two dimensions. These simulations are helpful to demonstrate the theoretical convergence of the algorithm but also demonstrate the exponential decay of the discrete energy over time. The two-dimensional examples further illustrate the behavior of displacement and thermal fields under specific boundary conditions, confirming the model's physical consistency.