This is addressed in the section "GK Dimension" here: https://arxiv.org/abs/1801.05369

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

This is addressed in the section "GK Dimension" of Ebrahim - The prime spectrum and representation theory of the 2×2 reflection equation algebra.

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