We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby, we obtain a number of positive characteristic stable invariants, such as the p-toral rank of HH¹(A, A). We also prove a more general result concerning Iwanaga–Gorenstein algebras, using a generalization of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory.