  We prove the cohomological version of the Sarnak-Xue Density Hypothesis for
$SO_5$ over a totally real field and for inner forms split at all finite
places. The proof relies on recent lines of work in the Langlands program: (i)
Arthur's Endoscopic Classification of Representations of classical groups, and
(ii) the Generalized Ramanujan-Petersson Theorem, proved for cohomological
self-dual cuspidal representations of general linear groups. We give
applications for the growth of cohomology of arithmetic manifolds,
density-Ramanujan complexes, cutoff phenomena and optimal strong approximation.
