I would like to try to calculate say $H_{\acute{e}tale}^*(B\mathbb{G}_{m, k}, \mathbb{Z}/p)$. Intuitively, since $B\mathbb{G}_m$ classifies line bundles, its topological analogue should be $\mathbb{CP}^{\infty}$, and so I expect the cohomology to be $\mathbb{Z}/p\mathbb{Z}[x], |x| = 2$, but i'm not entirely sure how to go about doing this calculation.
As a stack $B\mathbb{G}_m$ is given by $[*/\mathbb{G}_m]$, but it isn't clear to me how to provide an $\acute{e}$tale cover that would provide a reasonable Čech cohomology calculation.