  Geodesic balls in a simply connected space forms $\mathbb{S}^n$,
$\mathbb{R}^{n}$ or $\mathbb{H}^{n}$ are distinguished manifolds for comparison
in bounded Riemannian geometry. In this paper we show that they have the
maximum possible boundary volume among Miao-Tam critical metrics with connected
boundary provided that the boundary of the manifold is an Einstein
hypersurface. In the same spirit we also extend a rigidity theorem due to
Boucher et al. \cite{Bou} and Shen \cite{Shen} to $n$-dimensional static
metrics with positive constant scalar curvature, which provides another proof
of a partial answer to the Cosmic no-hair conjecture previously obtained by
Chru\'sciel \cite{Chrus}.
