AdS-Schwarzschild-like black hole thermodynamics: Loop quantum gravity impact on topology and universality

This study investigates the thermodynamic topology of AdS-Schwarzschild-like black holes within the framework of Loop Quantum Gravity (LQG) by employing various non-extensive entropy formulations. These include Rényi entropy, Barrow entropy, LQG entropy, and Sharma–Mittal entropy, each characterized by specific parameters that influence black hole thermodynamics. Our analysis reveals that adopting Rényi entropy instead of the traditional Bekenstein–Hawking entropy leads to significant changes in phase transition behavior, with a critical parameter value distinguishing between first-order and second-order transitions. While Barrow entropy remains consistent with the classical model across its parameter range, LQG entropy introduces complex topological features dependent on its defining parameter, including multiple zero points linked to distinct topological charges. Sharma–Mittal entropy exhibits a rich variety of topological structures that depend sensitively on the interplay between its two parameters, resulting in different patterns of topological charges. Furthermore, we confirm the existence of universal relations connecting entropy and extremality bounds for all entropy formulations studied, which remain valid upon introducing a small corrective term in the action. These findings highlight the profound impact of quantum gravity-inspired entropy modifications on the thermodynamic and topological properties of black holes.