  It is shown that the volume entropy of a Hilbert geometry associated to an
$n$-dimensional convex body of class $C^{1,1}$ equals $n-1$. To achieve this
result, a new projective invariant of convex bodies, similar to the
centro-affine area, is constructed. In the case $n=2$, and without any
assumption on the boundary, it is shown that the entropy is bounded above by
$\frac{2}{3-d} \leq 1$, where $d$ is the Minkowski dimension of the extremal
set of $K$. An example of a plane Hilbert geometry with entropy strictly
between 0 and 1 is constructed.
