Quantum corrected polymer black hole thermodynamics: mass relations and logarithmic entropy correction

In this paper, we continue the analysis of the effective model of quantum Schwarzschild black holes recently proposed by some of the authors in [1, 2]. In the resulting quantum-corrected spacetime the central singularity is resolved by a black-to-white hole bounce, quantum effects become relevant at a unique mass-independent curvature scale, while they become negligible in the low curvature region near the horizon and classical Schwarzschild geometry is approached asymptotically. This is the case independently of the relation between the black and white hole masses, which are thus freely specifiable independent observables. A natural question then arises about the phenomenological implications of the resulting non-singular effective spacetime and whether some specific relation between the masses can be singled out from a phenomenological perspective. Here we focus on the thermodynamic properties of the effective polymer black hole and analyze the corresponding quantum corrections as functions of black and white hole masses. The study of the relevant thermodynamic quantities such as temperature, specific heat, and horizon entropy reveals that the effective spacetime generically admits an extremal minimal-sized configuration of quantum-gravitational nature characterized by vanishing temperature and entropy. For large masses, the classically expected results are recovered at leading order and quantum corrections are negligible, thus providing us with a further consistency check of the model. The explicit form of the corrections depends on the specific relationship among the masses. In particular, a first-order logarithmic correction to the black hole entropy is obtained for a quadratic mass relation. The latter corresponds to the case of proper finite-length effects which turn out to be compatible with a minimal length generalized uncertainty principle associated with an extremal Planck-sized black hole.