A Fano variety $X$ is a (smooth) projective algebraic variety whose anticanonical bundle $-K_X$ is ample.
In characteristic $0$, by Kodaira vanishing, $h^i(X, O_X) = 0$ for all $i>0$.
In his paper "Fano varieties in positive characteristic", Shepherd-Barron proves that it's also true for Fano threefolds in positive characteristic.
In "Regular Del Pezzo surfaces with irregularity", Maddock constructs a regular (non smooth) Del Pezzo surface over an imperfect field of characteristic 2 having $h^1(X, O_X) \neq 0$.
Are there (other) examples of Fano varieties such that $h^i(X, O_X) \neq 0$ for some $i>0$ ? Ideally smooth varieties in dimension $d \geq 3$ over a perfect field of characteristic $p \geq 3$, but I'd be also happy if some of those conditions are not satisfied !