  Recently, Malle and Navarro obtained a Galois strengthening of Brauer's
height zero conjecture for principal $p$-blocks when $p=2$, considering a
particular Galois automorphism of order~$2$. In this paper, for any prime $p$
we consider a certain elementary abelian $p$-subgroup of the absolute Galois
group and propose a Galois version of Brauer's height zero conjecture for
principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when
$G$ does not involve certain groups of Lie type of small rank as composition
factors. Furthermore, we prove it for almost simple groups and for $p$-solvable
groups.
