Let $G$ be a reductive algebraic group over $\mathbb{C}$ acting on a scheme $X/\mathbb{C}$ of finite type. Let $R =\Gamma(X,\mathcal{O}_X)$ be the ring of global sections of $X$ and denote by $R^G$ the ring of invariants. Is $R^G$ a finite type algebra over $\mathbb{C}$?
I know this holds at least when $X$ is affine. If the result is negative in general, could you provide a counter-example?
Edit: As noticed in the comments, the result does not hold if $R$ is not at least noetherian. For the sake of the question, let us add this further assumption on $R$.