Zariski cohomology of $\mathbb{A}^1$ over a local ring with values in $\mathbb{G}_m$

Question

Let $X$ be a the spectrum of a regular local ring. What is known about the vanishing of the Zariski cohomology group $$ H^n(\mathbb{A}^k_X,\mathbb{G}_m) $$ for $n,k\geq 0$?

If $X$ has dimension $d$ and if $n>k+d$ so that $\mathbb{A}^k_X$ has dimension smaller than $n$, then the group is zero but can one say more?