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.