  We classify hypersurfaces with rotational symmetry and positive constant
$r$-th mean curvature in $\mathbb H^n \times \mathbb R$. Specific constant
higher order mean curvature hypersurfaces invariant under hyperbolic
translation are also treated. Some of these invariant hypersurfaces are
employed as barriers to prove a Ros--Rosenberg type theorem in $\mathbb H^n
\times \mathbb R$: we show that compact connected hypersurfaces of constant
$r$-th mean curvature embedded in $\mathbb H^n \times [0,\infty)$ with boundary
in the slice $\mathbb H^n \times \{0\}$ are topological disks under suitable
assumptions.
