  We study the asymptotic distribution of resonances for scattering by
compactly supported potentials in hyperbolic space. We first establish an upper
bound for the resonance counting function that depends only on the dimension
and the support of the potential. We then establish the sharpness of this
estimate by proving the a Weyl law for the resonance counting function holds in
the case of radial potentials vanishing to some finite order at the edge of the
support. As an application of the existence of potentials that saturate the
upper bound, we derive additional resonance asymptotics that hold in a suitable
generic sense. These generic results include asymptotics for the resonance
count in sectors.
