We classify those rotationally invariant translators of the mean curvature flow in the Hyperbolic Einstein Space-time \(\mathbb{H}^n\times_{-1}\mathbb{R}\). Next, we consider a connected, compact space-like translator whose boundary is the boundary of a bounded open domain in a slice. If the domain is invariant by an isometry \(\sigma\) of \(\mathbb{H}^n\), then the traslator is invariant by \(\sigma\times id\). We then characterize one of the rotationally invariant examples.