The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails

We show that the Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails for threefolds in any positive characteristic, and for terminal 3-folds in characteristic $p \in \{2, 3, 5\}$. To prove this, we introduce the notion of $\mathbb{F}_p$-rationality for singularities in positive characteristic and we show that klt singularities in dimension at most 4 are $\mathbb{F}_p$-rational. We apply this to prove a Frobenius--stable version of the Kawamata--Viehweg vanishing theorem on $K$-trivial 3-folds.