* Finally, I do not think it is possible for G to vary in the way that the author proposes. For one, we have constraints on the variation of G on Earth from the Oklo natural nuclear reactor and, going back further into the past, from Big Bang Nucleosynthesis which mean that if cannot have change by more than a few percent. If it were changing together with Hubble, we would find that the galaxies and orbits destabilise etc. Moreover, as the author notices, stars are very sensitive to the strength of gravity (e.g.) and we would have know about such variations.
The rest of the paper in fact makes use of some simple geometrical arguments to obtain distances and I have to admit I do not understand what they refer to. Proper null geodesics of the metric must be calculated to be able to say anything about distances.
* I do not understand why the author uses the standard equation for luminosity distance in LambdaCDM cosmology (calling it the "improved Hubble law"). This fundamentally assume an FRW geometry in 3 spatial dimensions, with the cosmological constant and non-relativitistic matter as the only sources of energy in the universe. If the author would like to make claims about how well supernovae fit his model, he should start with a metric and with a definition of how light propagates (e.g. on null geodesics in standard physics). Given the metric and its dynamics which one would obtain by solving Einstein equations, an equation for the luminosity distance-redshift relation can be derived. It would be different.
* The recent detection of gravitational waves by LIGO confirms that we understand reasonably well the generation and propagation of these waves on cosmological distances. For the reason explained above, they would also have been sensitive to the full 5D structure of the space time and therefore their luminosity would have been completely different in such a 5D setup.
* Even if one were to brush this off as some sort of new physics (e.g. inspired by open string theory), gravity must know about the extra dimensions. Again, we know with excellent precision that the force of gravity in the solar system falls off as 1/r^2 and therefore it is 3+1 dimensional. It is possible to have a model of gravity which is 4D at small distances but sensitive to the full 5D space at large distances (see the Dvali-Gabadadze-Porrati model) but this means that the universe behaves in a completely standard way until the acceleration era. This is not what the author has in mind and comes with its own inconsistencies (ghosts etc) which have not been solved.