Interpreting the cosmological constant as a pressure, whose thermodynamically conjugate variable is a volume, modifies the first law of black hole thermodynamics. Properties of the resulting thermodynamic volume are investigated: the compressibility and the speed of sound of the black hole are derived in the case of nonpositive cosmological constant. The adiabatic compressibility vanishes for a nonrotating black hole and is maximal in the extremal case—comparable with, but still less than, that of a cold neutron star. A speed of sound $v_s$ is associated with the adiabatic compressibility, which is equal to $c$ for a nonrotating black hole and decreases as the angular momentum is increased. An extremal black hole has $v_s^2=0.9c^2$ when the cosmological constant vanishes, and more generally $v_s$ is bounded below by $c/\sqrt{2}$.

• Received 2 September 2011

DOI:https://doi.org/10.1103/PhysRevD.84.127503