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

TL;DR

This paper explores charged black holes in massive gravity's rainbow, analyzing their thermodynamics, stability, and phase transitions, revealing van der Waals-like behavior across different horizon topologies using various approaches.

Contribution

It introduces exact solutions for topological charged black holes in energy-dependent massive gravity and studies their thermodynamics and phase transitions with novel methods.

Findings

Black holes exhibit van der Waals-like phase transitions.

Thermodynamic stability depends on parameters and horizon topology.

Critical values are obtained in extended phase space.

Abstract

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.