This is very basic and appears throughout numerous applications, but by Serre vanishing...
For ample $L$, Frobenius eventually kills cohomology of $H^i(X, L^{-1})$ for $i < \dim X$ and kills $H^i(X, L)$ for $i > 0$.
(In other words, the natural map $F^e : H^i(X, L) \to H^i(X, L^{p^e})$ has zero target for $e \gg 0$.)
