  The primal cohomology $\mathbb{K}_\mathbb{Q}$ of the theta divisor $\Theta$
of a principally polarized abelian fivefold (ppav) is the direct sum of its
invariant and anti-invariant parts $\mathbb{K}_\mathbb{Q}^{+1}$, resp.
$\mathbb{K}_\mathbb{Q}^{-1}$ under the action of $-1$. For smooth $\Theta$,
these have dimension $6$ and $72$ respectively. We show that
$\mathbb{K}_\mathbb{Q}^{+1}$ consists of Hodge classes and, for a very general
ppav, $\mathbb{K}_\mathbb{Q}^{-1}$ is a simple Hodge structure of level $2$.
