  Building on a technical result by Brunnemann and Rideout on the spectrum of
the Volume operator in Loop Quantum Gravity, we show that the dimension of the
space of the quadrivalent, diffeomorphism invariant states with no zero-volume
nodes describing a region with total volume smaller than $V$, has \emph{finite}
dimension, bounded by $V \log V$. This allows us to introduce the notion of
"volume entropy" for this phase space: the von Neumann entropy associated to
the measurement of volume.
