  We prove the spacetime Penrose inequality for asymptotically flat
$2(n+1)$-dimensional initial data sets for the Einstein equations, which are
invariant under a cohomogeneity one action of $\mathrm{SU}(n+1)$. Analogous
results are obtained for asymptotically hyperbolic initial data that arise as
spatial hypersurfaces in asymptotically Anti de-Sitter spacetimes. More
precisely, it is shown that with the dominant energy condition, the total mass
is bounded below by an explicit function of the outermost apparent horizon
area. Furthermore, the inequality is saturated if and only if the initial data
isometrically embed into a Schwarzschild(-AdS) spacetime. This generalizes the
only previously known case of the conjectured spacetime Penrose inequality,
established under the assumption of spherical symmetry. Additionally, in the
time symmetric case, we observe that the inequality holds for
$4(n+1)$-dimensional and 16-dimensional initial data invariant under
cohomogeneity one actions of $\mathrm{Sp}(n+1)$ and $\mathrm{Spin}(9)$,
respectively, thus treating the inequality for all cohomogeneity one actions in
this regime.
