On variations of $G$ in the geometric scalar theory of gravity

We analyze the possible variability of the effective Newtonian gravitational constant $G_{\mathrm{N}}$ in space and time in the framework of the geometric scalar theory of gravity suggested by M. Novello et al. [JCAP 06, 014 (2013); arXiv: 1212.0770]. Spatial variations of $G_{\mathrm{N}}$ in the Solar system are shown to have orders of magnitude detectable by modern instruments. As to variations of $G_{\mathrm{N}}$ with cosmological time, it is shown (at least for the particular formulation of the theory discussed in the original paper and the corresponding cosmological models) that these variations are more rapid than is allowed by observations.