  We prove that the degree $r(2p-3)$ cohomology of any (untwisted) finite group
of Lie type over $\mathbb{F}_{p^r}$, with coefficients in characteristic $p$,
is nonzero as long as its Coxeter number is at most $p$. We do this by
providing a simple explicit construction of a nonzero element. Furthermore, for
the groups $GL_n\mathbb{F}_{p^r}$, when $p=2$ or $r=1$, we calculate the
cohomology in degree $r(2p-3)$, for all $n$.
