The scalar torus rigidity theorem says that a Riemannian manifold, that is diffeomorphic to a torus and has nonnegative scalar curvature, is isometric to a flat torus. In this talk I adress the corresponding stability question. The main theorem states that for a class of graphical Riemannian tori almost nonnegative scalar curvature implies closeness to a flat torus w.r.t. the Sormani-Wenger intrinsic flat topology. This is joint work with Armando J. Cabrera Pacheco and Raquel Perales.