  Viewing the cosmological constant $\Lambda<0$ 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 $\Lambda$ as independent variable on the same
footing as the quasilocal energy approach.
