  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 $\mathrm{HH}^1(A,A)$. We also prove a
more general result concerning Iwanaga-Gorenstein algebras, using a more
general notion of stable equivalences of Morita type. Several applications are
given to commutative algebra and modular representation theory. These results
are proven by first establishing the stable invariance of the
$B_\infty$-structure of the Hochschild cochain complex. In the appendix we
explain how the $p$-power operation on Hochschild cohomology can be seen as an
artifact of this $B_\infty$-structure. In particular, we establish
well-definedness of the $p$-power operation, following some -- originally
topological -- methods due to May, Cohen and Turchin, using the language of
operads.
