It is even a derived invariant. Here is a proof in a special case, but Im not sure whether it works more general (with the same proof?) Let $A$ and $B$ two noetherian $K$-algebras for a commutative ring $K$ that are projective as $K$-modules. Then a standard derived equivalence $F=X \otimes_A^L: D^b(A) \rightarrow D^b(B)$ with quasi-inverse $G=Y \otimes_B^L $ induces a derived equivalence between the derived bimodule categories $H=(X \otimes_A^L - ) \otimes_A^L Y : D^b(A^e) \rightarrow D^b(B^e)$ where $A^e=A^{op} \otimes_K A$.

Now the Hochschild cohomology has terms $Ext_{A^e}^l(A,A)$ but $H$ sends $A$ to $B$ and thus preserved Hochschild cohomology. (it also preserved $A^{*}$ in case $A$ is finite dimensional, not sure about the general case)

It is even a derived invariant. Here is a proof in a special case, but Im not sure whether it works more general (with the same proof?) Let $A$ and $B$ two noetherian $K$-algebras for a commutative ring $K$ that are projective as $K$-modules. Then a standard derived equivalence $F=X \otimes_A^L: D^b(A) \rightarrow D^b(B)$ with quasi-inverse $G=Y \otimes_B^L $ induces a derived equivalence between the derived bimodule categories $H=(X \otimes_A^L - ) \otimes_A^L Y : D^b(A^e) \rightarrow D^b(B^e)$ where $A^e=A^{op} \otimes_K A$.

Now the Hochschild cohomology has terms $Ext_{A^e}^l(A,A)$ but $H$ sends $A$ to $B$ and thus preserves Hochschild cohomology. (it also preserves $A^{*}$ in case $A$ is finite dimensional, not sure about the general case)