Thermodynamics of the quantum Schwarzschild black hole

We discuss some thermodynamic properties as well as the stability of a quantum Schwarzschild black hole, comparing the results with those obtained within a bumblebee gravity model. In particular, the Hawking temperature, T H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{{\text{H}}}$$\end{document} , the entropy, S, the heat capacity, C, and the Gibbs free energy, G, are computed for both cases. In addition to that, we compute the Brown-York quasilocal energy and compare the solution with the Schwarzschild case. We find that in both cases (quantum Schwarzschild and bumblebee gravity model) the temperature, the entropy, and the heat capacity show the same functional form, under the replacement lambda 2 -> & ell; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda <^>2 \rightarrow \ell$$\end{document} and vice versa. Specifically, the temperature is found to be lower compared to the classical (Schwarzschild) solution; whereas, the entropy is computed to be larger. Moreover, the heat capacity becomes more negative. Notably, a distinct contrast emerges in obtaining the Gibbs free energy between these two cases, and this distinction appears to stem from the ADM mass.