on the nonlinear equation of state in black hole thermodynamics

The study of black hole thermodynamics has revealed profound connections between gravitation, quantum theory, and statistical mechanics. in many instances, the key physical information is encoded in nonlinear algebraic or transcendental equations that relate horizon radius, temperature, and pressure. in this work, we examine a specific nonlinear equation arising from the extended phase space of charged anti-de sitter (ads) black holes. by analyzing its structure and obtaining approximate and exact solutions, we highlight the physical implications for the thermodynamic stability of black holes. our results clarify the role of nonlinearities in determining critical points and phase transitions analogous to the van der waals fluid.