Beyond entropy extension and exploring the thermodynamics of AdS black holes

We investigate the thermodynamic properties of various Anti-de Sitter (AdS) black holes by extending the conventional Bekenstein–Hawking entropy to the non-extensive frameworks of Tsallis and Barrow. Focusing on charged, rotating, and Gauss–Bonnet AdS black holes, we compute fundamental thermodynamic quantities such as mass, temperature, specific heat, Gibbs free energy, and the Joule–Thomson coefficient. Our analysis shows that charged AdS black holes remain thermodynamically unstable in both extensive and non-extensive descriptions, while rotating and Gauss–Bonnet AdS black holes exhibit improved stability under generalized entropy formalisms. We further demonstrate that the Tsallis and Barrow entropies significantly modify phase transition structures, inversion curves, and heating–cooling regimes, thereby providing a richer picture of black hole thermodynamics. Within the holographic framework, these modifications suggest that the dual conformal field theories depart from Gibbs statistics and may encode long-range interactions or fractal-like correlations. The results highlight the role of generalized entropies in uncovering novel stability features and in offering deeper insights into black hole microstructure, holographic duality, and quantum gravity phenomenology.