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Violation of Lorentz invariancy in the high energy quantum gravity motivatesone to consider an energy dependent spacetime with massive deformation ofstandard general relativity. In this paper, we take into account an energydependent metric in the context of a massive gravity model to obtain exactsolutions. We investigate the geometry of black hole solutions and alsocalculate the conserved and thermodynamic quantities, which are fullyreproduced by the analysis performed with the standard techniques. Afterexamining the validity of the first law of thermodynamics, we conduct a studyregarding the effects of different parameters on thermal stability of thesolutions. In addition, we employ the relation between cosmological constantand thermodynamical pressure to study the possibility of phase transition.Interestingly, we will show that for the specific configuration considered inthis paper, van der Waals like behavior is observed for different topology. Inother words, for flat and hyperbolic horizons, similar to spherical horizon, asecond order phase transition and van der Waals like behavior are observed.Furthermore, we use geometrical method to construct phase space and study phasetransition and bound points for these black holes. Finally, we obtain criticalvalues in extended phase space through the use of a new method.

Topological charged black holes in massive gravity's rainbow and their thermodynamical analysis through various approaches