Thermodynamic metric usually works only for those black holes with more than one conserved charge, thereby excluding the Schwarzschild black hole. In this paper, however, different versions of thermodynamic metric are computed and compared for the Schwarzschild-like black hole by introducing new degrees of freedom. These new degrees of freedom have two purposes. First, the deformed metric may be treated offshell to the ordinary Schwarzschild black hole, and onshell physics corresponds to the submanifold by gauge fixing of this additional degree of freedom. In particular, the thermal Ricci scalar for the Schwarzschild black hole, though different for various deformations, can be obtained by switching off the deformation. Second, while deformed metric is treated onshell, a divergent Ricci scalar may signal an exotic phase in which new physical degrees of freedom manifest. This paper considers the new degree of freedom as the running Newton constant, a cutoff scale for regular black holes, a noncommutative deformation or the deformed parameter in the nonextensive Tsallis–Rènyi entropy.
Black hole scalarization with Gauss-Bonnet and Ricci scalar couplings
