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We consider the thermodynamics of rotating and charged asymptotically deSitter black holes. Using Hamiltonian perturbation theory techniques, we derivethree different first law relations including variations in the cosmologicalconstant, and associated Smarr formulas that are satisfied by such spacetimes.Each first law introduces a different thermodynamic volume conjugate to thecosmological constant. We examine the relation between these thermodynamicvolumes and associated geometric volumes in a number of examples, includingKerr-dS black holes in all dimensions and Kerr-Newman-dS black holes in D=4. Wealso show that the Chong-Cvetic-Lu-Pope solution of D=5 minimal supergravity,analytically continued to positive cosmological constant, describes black holesolutions of the Einstein-Chern-Simons theory and include such chargedasymptotically de Sitter black holes in our analysis. In all these examples wefind that the particular thermodynamic volume associated with the regionbetween the black hole and cosmological horizons is equal to the naivegeometric volume. Isoperimetric inequalities, which hold in the examplesconsidered, are formulated for the different thermodynamic volumes andconjectured to remain valid for all asymptotically de Sitter black holes. Inparticular, in all examples considered, we find that for fixed volume of theobservable universe, the entropy is increased by adding black holes. Weconjecture that this is true in general.

Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes