  In this work we study the Thermodynamics of D-dimensional Schwarzschild-anti
de Sitter (SAdS) black holes. The minimal Thermodynamics of the SAdS spacetime
is briefly discussed, highlighting some of its strong points and shortcomings.
The minimal SAdS Thermodynamics is extended within a Hamiltonian approach, by
means of the introduction of an additional degree of freedom. We demonstrate
that the cosmological constant can be introduced in the thermodynamic
description of the SAdS black hole with a canonical transformation of the
Schwarzschild problem, closely related to the introduction of an anti-de Sitter
thermodynamic volume. The treatment presented is consistent, in the sense that
it is compatible with the introduction of new thermodynamic potentials, and
respects the laws of black hole Thermodynamics. By demanding homogeneity of the
thermodynamic variables, we are able to construct a new equation of state that
completely characterizes the Thermodynamics of SAdS black holes. The treatment
naturally generates phenomenological constants that can be associated with
different boundary conditions in underlying microscopic theories. A whole new
set of phenomena can be expected from the proposed generalization of SAdS
Thermodynamics.
