  Let $\mathbb{H}^{n+1}_1$ be the $(n+1)$-dimensional anti-de Sitter space, in
this paper we propose to extend $\mathbb{H}^{n+1}_1$ conformally to another
copy of $\mathbb{H}^{n+1}_1$ by gluing them together along the boundary at
infinity, and denote the resulting space by \emph{double anti-de Sitter space}
$\mathbb{DH}^{n+1}_1$. We introduce a volume (possibly complex valued) on
polytopes in $\mathbb{DH}^{n+1}_1$ whose facets all have non-degenerate metrics
(called \emph{good} polytopes), and show that the volume is well defined and
invariant under isometry, including the case that the good polytope contains a
non-trivial portion of $\partial\mathbb{H}^{n+1}_1$. For $n$ even, the volume
of a good polytope in $\mathbb{DH}^{n+1}_1$ is shown to be completely
determined by its intersection with $\partial\mathbb{H}^{n+1}_1$, which leads
to the following main consequences: it induces a new intrinsic \emph{volume} on
$\partial\mathbb{H}^{n+1}_1$ that is invariant under conformal transformations,
and establishes an AdS-CFT type correspondence between the volumes on
$\mathbb{DH}^{n+1}_1$ and $\partial\mathbb{H}^{n+1}_1$.
