  Treating the cosmological constant as a pressure, in the context of black
hole thermodynamics, a thermodynamic volume for the black hole can be defined
as being the thermodynamic variable conjugate to the pressure, in the sense of
a Legendre transform. The thermodynamic volume is explicitly calculated, as the
Legendre transform of the pressure in the enthalpy for a rotating
asymptotically anti-de Sitter Myers-Perry black hole in D space-time
dimensions. The volume obtained is shown to agree with previous calculations
using the Smarr relation. The compressibility is calculated and shown to be
non-negative and bounded.
  Taking the limit of zero cosmological constant, the compressibility of a
rotating black hole in asymptotically flat space-times is determined and the
corresponding speed of sound computed. The latter is bounded above and has an
elegant expression purely in terms of the angular momenta, in the form of
quartic and quadratic Casimirs of the rotation group, SO(D-1).
