Quantum corrections to the thermodynamics and phase transition of a black hole surrounded by a cavity in the extended phase space

Abstract: In the extended phase space, we investigate the rainbow gravity-corrected thermodynamic phenomena and phase structure of the Schwarzschild black hole surrounded by a spherical cavity. The results show that rainbow gravity has a very significant effect on the thermodynamic phenomena and phase structure of the black hole. It prevents the black hole from total evaporation and leads to a remnant with a limited temperature but no mass. Additionally, we restore the $P-V$ criticality and obtaine the critical quantities of the canonical ensemble. When the temperature or pressure is smaller than the critical quantities, the system undergoes two Hawking-Page-like phase transitions and one first-order phase transition, which never occurs in the original case. Remarkably, our findings demonstrate that the thermodynamic behavior and phase transition of the rainbow SC black hole surrounded by a cavity in the extended phase space are analogous to those of the Reissner-Nordstr\"{o}m anti-de Sitter black hole. Therefore, rainbow gravity activates the effect of electric charge and cutoff factor in the evolution of the black hole.