Quantum-corrected black hole thermodynamics from the gravitational path integral

Exploring quantum effects from black hole thermodynamics has always been a pivotal topic. In recent years, the free energy landscape and ensemble-averaged theory based on the Euclidean path integral approach have provided further understanding of the statistical aspects of the black hole system. We investigate the quantum-corrected thermodynamics of the Reissner-Nordstrom-AdS black hole by including off-shell geometries in the path integral. We obtain a one-loop effective action by considering the subleading-order terms in the ensemble-averaged theory, and verify that the effective thermodynamic quantities consistent with the effective action define a valid thermodynamics. Furthermore, the phase diagram was modified by the off-shell effects, resulting in a more abundant phase structure. We discover that the traditional black hole thermodynamics can be recovered in the semi-classical limit. The region of first-order phase transitions shrinks and zero-order phase transitions emerge when off-shell effects are included. The results provide new perspectives for understanding the quantum corrections in black hole thermodynamics.