Let $G$ be a reductive linear algebraic group over a field $k$ . Let $A$ be a finitely generated commutative $k$ -algebra on which $G$ acts rationally by $k$ -algebra automorphisms. Invariant theory states that the ring of invariants $A^G=H^0(G,A)$ is finitely generated. We show that in fact the full cohomology ring $H^*(G,A)$ is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed in [22]. We also continue the study of bifunctor cohomology of $\Gamma^*(\mathfrak{gl}^{(1)})$