  For certain compactly supported metric and/or potential perturbations of the
Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance
counting function with an explicit constant that depends only on the dimension,
the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of
the metric perturbation. This constant is shown to be sharp in the case of
scattering by a spherical obstacle.
