I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal torus $D_{r \pm 1}$ (where $|D_{2n}| = 2n$), dependent on which is divisible by $p$, but am struggling to find a reference for $\operatorname{H}^i(D_{2n}, \mathbb{F}_p)$ for $i > 2$ though this must almost certainly be known. 

I can see by computations in magma that the answer should be $0$ for $i \equiv 1, \, 2 \mod 4$ and $\mathbb{F}_p$ otherwise.