  We study the spectrum of the Finsler--Laplace operator for regular Hilbert
geometries, defined by convex sets with $C^2$ boundaries. We show that for an
$n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$,
which we prove to be the infimum of the essential spectrum. We also construct
examples of convex sets with arbitrarily small eigenvalues.
