  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.

High Energy Physics - Theory;General Relativity and Quantum Cosmology