On the total volume of the double hyperbolic space

Let the \emph{double hyperbolic space} ${\mathbb{D}\mathbb{H}}^n$, proposed in this paper as an extension of the hyperbolic space ${\mathbb{H}}^n$, contain a two-sheeted hyperboloid with the two sheets connected to each other along the boundary at infinity. We propose to extend the volume of convex polytopes in ${\mathbb{H}}^n$ to polytopes in ${\mathbb{D}\mathbb{H}}^n$, where the volume is invariant under isometry but can possibly be complex valued. We show that the total volume of ${\mathbb{D}\mathbb{H}}^n$ is equal to $i^n{V}_n({\mathbb{S}}^n)$ for both even and odd dimensions, and prove a Schl"{a}fli differential formula (\SDF{}) for ${\mathbb{D}\mathbb{H}}^n$. For $n$ odd, the volume of a polytope in ${\mathbb{D}\mathbb{H}}^n$ is shown to be completely determined by its intersection with $\mathrm{\partial }{\mathbb{H}}^n$ and induces a new intrinsic \emph{volume} on $\mathrm{\partial }{\mathbb{H}}^n$ that is invariant under M"{o}bius transformations.