We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, with respect to perturbations in Lp∩L∞ for any p∈(1,n), improving a result by Deruelle and the first author [15] . Our result applies to all ALE gravitational instantons. The theorem is proved by a fixed point argument, based on novel estimates for the heat kernel of the Lichnerowicz Laplacian. It allows us to give a precise description of the convergence behaviour of the Ricci flow. Our decay rates are strong enough to prove positive scalar curvature rigidity in Lp, for each p∈[1,nn−2), generalizing a result by Appleton.