  In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local
cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible
submodules. This answers positively an open question raised by F.Enescu and
M.Hochster. We also prove that if $(R,\m)$ is excellent and is $F$-pure on the
punctured spectrum, then all local cohomology modules have finite length in the
category of $R$-modules with Frobenius action. Finally, we show that the
property that all $H_{\m}^(R)$ have finitely many Frobenius compatible
submodules passes to localizations.
