Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that: $$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$
Well if the manifold is $\mathbb R ^n$ and $F$ only depends on the (geodesic) distance of two given points in $\mathbb R^n$ one can apply the divergence theorem considering the volume of the sphere centered in one of the points, whose radius, $R$, is the distance between the considered points, and $\partial U$ is just the sphere's surface. Then the divergence theorem can be writen explicitly as
$$\int_{S^n}\nabla \cdot F \space \space R^n\sin^{n-1}(\theta)d\theta d\Omega=F \space R^n \int_{ S^{n-1}} \sin^{n-1}(\theta)d\theta d\Omega .$$
But what if the considered manifold is the hyperbolic space? What would be the volume element? $R^n\sinh^{n-1}(\theta)d\theta d\Omega$, as the wiki article suggests? If so, what would be interpretation of $R$ and the angles?