Abstract: We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with ''controlled geometry '') and such that their linear part is hyperbolic.

In absence of the simplicity condition it is possible to  construct a robustly transitive counter-example, evidencing the necessity of our assumptions.

Work in progress joint to Pablo Carrasco (UFMG-Brazil),  Cristina Lizana (UFBA -Brazil) and  E. Pujals (CUNY, USA).