Black hole thermodynamics with the cosmological constant as independent variable: Bridge between the enthalpy and the Euclidean path integral approaches

Viewing the cosmological constant Λ<0 as an independent variable, we consider the thermodynamics of the Schwarzschild black hole in an anti-de Sitter (AdS) background. For this system, there is one approach which regards the enthalpy as the master thermodynamic variable and makes sense if one considers the vacuum pressure due to the cosmological constant acting in the volume inside the horizon and the outer size of the system is not restricted. From this approach a first law of thermodynamics emerges naturally. There is yet another approach based on the Euclidean action principle and its path integral that puts the black hole inside a cavity, defines a quasilocal energy at the cavity's boundary, and from which a first law of thermodynamics in a different version also emerges naturally. The first approach has affinities with critical phenomena in condensed matter physics and the second approach is an ingredient necessary for the construction of quantum gravity. The bridge between the two approaches is carried out rigorously, putting thus the enthalpic thermodynamics with Λ as independent variable on the same footing as the quasilocal energy approach.