The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the eigenvalues of a random element of an associated compact group. There are conjectural generalizations for curves of higher genus and abelian varieties of higher dimension in terms of more complicated compact groups. 

> What's conjectured about the corresponding question for hypersurfaces?

To be more precise, consider a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ defined over $\mathbb{Z}$. Its only interesting $\ell$-adic cohomology should be in the middle degree, and its rank is known. What's conjectured about the distribution of the eigenvalues of Frobenius at $p$ acting on it as $p$ varies? 

**Edit:** If, as ulrich suggests, the expectation here is that it resembles the case of curves / abelian varieties, what about the case of more general varieties? In general it seems like the cup product restricts Frobenius eigenvalues in some a priori complicated way.