Variational Effects of the Gravitational Coupling <i>G</i> on the Thermodynamics of a Charged Reissner–Nordström–AdS Black Hole

This study delves into the thermodynamic characteristics of a Reissner–Nordström-AdS black hole characterized by electric charge (Q). By treating the black hole as a thermodynamic system characterized by a well-defined temperature and entropy, the paper explores its dynamical behavior. It focuses on how Variable gravitational coupling influences these thermodynamic quantities, with particular attention to black hole horizons, the mass function, entropy, surface gravity and Hawking temperature.This study investigates the P-V critical behavior of the given black hole by examining its isotherms and comparing them with those characteristic of a van der Waals fluid. This study also emphasizes that the transition between small and large black holes parallels a liquid-gas phase shift, with first-order and second-order phase changes analogous to fluid behaviors. The gravitational coupling G controls the Hawking temperature, critical point location, entropy, and charge bound in charged AdS black holes. Larger G suppresses temperature and shifts phase transitions to smaller scales, while weaker gravity allows larger charge and entropy. These results reveal significant deviations from the usual Schwarzschild geometry and emphasize how electric charge, curvature corrections, and a modified gravitational coupling together determine the causal structure of the black hole spacetime.