  Violation of Lorentz invariancy in the high energy quantum gravity motivates
one to consider an energy dependent spacetime with massive deformation of
standard general relativity. In this paper, we take into account an energy
dependent metric in the context of a massive gravity model to obtain exact
solutions. We investigate the geometry of black hole solutions and also
calculate the conserved and thermodynamic quantities, which are fully
reproduced by the analysis performed with the standard techniques. After
examining the validity of the first law of thermodynamics, we conduct a study
regarding the effects of different parameters on thermal stability of the
solutions. In addition, we employ the relation between cosmological constant
and thermodynamical pressure to study the possibility of phase transition.
Interestingly, we will show that for the specific configuration considered in
this paper, van der Waals like behavior is observed for different topology. In
other words, for flat and hyperbolic horizons, similar to spherical horizon, a
second order phase transition and van der Waals like behavior are observed.
Furthermore, we use geometrical method to construct phase space and study phase
transition and bound points for these black holes. Finally, we obtain critical
values in extended phase space through the use of a new method.
