  The action and the thermodynamics of a rotating black hole in the presence of
a positive cosmological constant are analyzed. Since there is no spatial
infinity, one must bring in, instead, a platform where the parameters
characterizing the thermodynamic ensemble are specified. In the present
treatment the platform in question is taken to be one of the two horizons,
which is considered as a boundary. If the boundary is taken to be the
cosmological horizon one deals with the action and thermodynamics of the black
hole horizon. Conversely, if one takes the black hole horizon as the boundary,
one deals with the action and thermodynamics of the cosmological horizon. The
two systems are different. Their energy and angular momenta are equal in
magnitude but have opposite sign. In either case, the energy and the angular
momentum are obtained as surface terms on the boundary, according to the
standard Hamiltonian procedure. The temperature and the rotational chemical
potential are also expressed in terms of magnitudes on the boundary. If, in the
resulting expressions, one continues the cosmological constant to negative
values, the black hole thermodynamic parameters defined on the cosmological
horizon coincide with those calculated at spatial infinity in the
asymptotically anti-de Sitter case.
