If $k$ has prime characteristic dividing $|G|$ then there are natural examples where $\operatorname{gldim}(A^G)=\infty$.

For example, let $A=\operatorname{End}_k(kG)$ be the ring of linear endomorphisms of the regular $kG$-module, which is a matrix ring over $k$ and so has global dimension zero, and let $G$ act by conjugation. Then $A^G=\operatorname{End}_{kG}(kG)\cong kG$, which has infinite global dimension.