$GL_n(\mathbb F_q)$ naturally acts on the vector space $V=\mathbb F_q^n$. As $GL_n(\mathbb F_q)$ is a finite group, the cohomology group $H^i(GL_n(\mathbb F_q),V)$ are all finite abelian groups. Can we compute those cohomology groups explicitly? 

This is a baby example, and for odd $p$ one can use the trick in https://mathoverflow.net/questions/187486/cohomology-of-sl2-r-with-coefficients-given-by-linear-action/187500#187500. In general, Let $G=\mathbb G(\mathbb F_q)$ where $\mathbb G$ is a connected reductive group over $\mathbb F_q$ (or more generally a finite group of Lie type), $V$ be an irreducible algebraic representation of $\mathbb G$ defined over $\mathbb F_{q^n}$ (or more generally any irreducible equal characteristic modular representation), can we compute $H^i(G,V)$ explicitly or at least give some bounds?