Gibbons-Hawking action for electrically charged black holes in the canonical ensemble and Davies' thermodynamic theory of black holes

We establish the connection between the Gibbons-Hawking Euclidean path integral approach applied to the canonical ensemble of a Reissner-Nordström black hole and the thermodynamic theory of black holes of Davies. We build the ensemble, characterized by a reservoir at infinity at temperature $T$ and electric charge $Q$, in $d$ dimensions. The Euclidean path integral yields the action and partition function. In zero loop, we uncover two solutions, one with horizon radius $r_{+1}$ the least massive, the other with $r_{+2}$, both meeting at a saddle point with radius $r_{+s}$ at temperature $T_s$. We derive the thermodynamics, finding that the heat capacity diverges at the turning point $T_s$ for each solution. The free energy of the stable solution is positive, so if the system is a black hole it makes a transition to hot flat space with charge at infinity. For a given $Q$ and $T>T_s$, there is only hot space. An interpretation of the results as energy wavelengths is attempted. For $d=4$, the thermodynamics from the path integral applied to the canonical ensemble is precisely the Davies thermodynamics theory of black holes, with $T_s$ being the Davies point. We sketch the case $d=5$.