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, except in 7 dimensions, a smooth limit exists where \(\Lambda\) or \(g\) goes to zero, providing a definition of \(V\) even for asymptotically-flat black holes.
