This is addressed in the section "GK Dimension" of [Ebrahim - The prime spectrum and representation theory of the 2×2 reflection equation algebra](https://arxiv.org/abs/1801.05369).

If the automorphism $\sigma$ is locally algebraic, then indeed $\operatorname{GKdim}(D(\sigma,a))=\operatorname{GKdim}(D)+1$.