Extended thermodynamical topology and phase structure of a quantum-corrected AdS black hole

In the framework of extended black hole thermodynamics, we study the thermodynamical topology and phase structure of a quantum-corrected AdS black hole. By introducing the minimal length scale as an additional thermodynamic variable, we derive the generalized first law and Smarr relation incorporating quantum corrections. The minimal length modifies the mass, temperature, and volume of the black hole while preserving the Bekenstein–Hawking area law of entropy at the semiclassical level. Using the extended thermodynamical topology method, we construct vector fields of different orders and determine the corresponding topological numbers to characterize phase stability. The results reveal the thermodynamic phase structure of the black hole, showing small unstable and large stable phases, with no higher-order critical points. Moreover, the analysis of the photon sphere reveals a single unstable photon orbit, confirming the genuine black hole nature of the spacetime.