  We prove well-posedness and rough path stability of a class of linear and
semi-linear rough PDE's on $\mathbb{R}^d$ using the variational approach. This
includes well-posedness of (possibly degenerate) linear rough PDE's in
$L^p(\mathbb{R}^d)$, and then -- based on a new method -- energy estimates for
non-degenerate linear rough PDE's. We accomplish this by controlling the energy
in a properly chosen weighted $L^2$-space, where the weight is given as a
solution of an associated backward equation. These estimates then allow us to
extend well-posedness for linear rough PDE's to semi-linear perturbations.
