Let $X$ be a smooth projective rational variety over $mathbb{C}$, and let $pi:Yrightarrow X$ be a principal projective bundle with fibers isomorphic to $SL(n,mathbb{C})/P$, where $P$ is a parabolic subgroup. Consider the sheaf $mathbb{G}_m$ on $Y$ in e’tale topology.

I’m mainly interested in the cohomologies of the higher direct images. Can we describe the higher direct images $R^ipi_*(mathbb{G}_m)$ on $X$? For example, is $pi_*mathbb{G}_m$ a locally constant sheaf on $X$?

Any help would be appreciated.