In a theory where the cosmological constant \Lambda or the gauge coupling constant g arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda, where E is now the enthalpy of the spacetime, and \Theta, the thermodynamic conjugate of \Lambda, is proportional to an effective volume V = -\frac{16 \pi \Theta}{D-2} "inside the event horizon." Here we calculate \Theta and V for a wide variety of D-dimensional charged rotating asymptotically AdS black hole spacetimes, using the first law or the Smarr relation. We compare our expressions with those obtained by implementing a suggestion of Kastor, Ray and Traschen, involving Komar integrals and Killing potentials, which we construct from conformal Killing-Yano tensors. We conjecture that the volume V and the horizon area A satisfy the inequality R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1, where {\cal A}_{D-2} is the volume of the unit (D-2)-sphere, and we show that this is obeyed for a wide variety of black holes, and saturated for Schwarzschild-AdS. Intriguingly, this inequality is the "inverse" of the isoperimetric inequality for a volume V in Euclidean (D-1) space bounded by a surface of area A, for which R\le 1. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the entropy inside a horizon of a given "volume" V is maximised for Schwarzschild-AdS. The thermodynamic definition of V requires a cosmological constant (or gauge coupling constant). However a smooth limit exists where \Lambda or g goes to zero, providing a definition of V even for asymptotically-flat black holes.