  In the present work we study the construction of different thermodynamic
descriptions for the Kerr-anti-de Sitter (KadS) black holes. The early versions
of the KadS thermodynamics are briefly discussed, highlighting some of its
strong points and shortcomings. Isohomogeneous transformations, a procedure for
generating new thermodynamics, are presented and geometrically interpreted for
KadS. This tool is used to determine possible KadS thermodynamics that can be
constructed to satisfy a Smarr formula, and the validity of the first law in
the generated thermodynamics. The connection between new thermodynamic theories
and early Hawking's approach is considered. In this new framework, the usual
KadS thermodynamics is complemented with its geometric construction, and
Hawking's proposal, which does not satisfy the first law, is improved to an
alternative thermodynamic theory. With the quantum statistical relation,
Hawking's and this alternative KadS thermodynamics are also generalized from
four to higher dimensions.
