  We consider scattering by star-shaped obstacles in hyperbolic space and show
that resonances satisfy a universal bound $\mathrm{Im}\,\lambda \leq
-\frac{1}{2}$ which is optimal in dimension $2$. In odd dimensions we also show
that $\mathrm{Im}\,\lambda \leq -\frac{\mu}{\rho}$ for a universal constant
$\mu$, where $\rho$ is the radius of a ball containing the obstacle; this gives
an improvement for small obstacles. In dimensions $3$ and higher the proofs
follow the classical vector field approach of Morawetz, while in dimension $2$
we obtain our bound by working with spaces coming from general relativity. We
also show that in odd dimensions resonances of small obstacles are close, in a
suitable sense, to Euclidean resonances.
