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"Hubble constant" is a misnomer, and an old fashioned term. Cosmologist actually use the term "Hubble parameter" (which is in general time dependent except in very specific models which only contain dark energy), and denote its present value as H~0~.

https://en.wikipedia.org/wiki/Hubble%27s_law#Time-dependence_of_Hubble_parameter

What you're talking about is the energy dependence of the coupling constants, which is a phenomenon that is very well understood theoretically, and also checked in experiments. The early universe was much hotter, and thus particles had much more kinetic energy and "felt" slightly different coupling constants. The neat thing is that, since this is a purely energy-dependent effect, we can recreate the conditions of the early universe: the collisions at LHC have an energy of the order of 1 TeV, which corresponds to a temperature of 10^16^K, the temperature 10^-12^ s after the Big Bang. Anything after the first 10^-12^ s we can directly recreate, and from 10^-12^ s to about 10^-30^ s-ish we can more or less reliably extrapolate. And of course this is all included in the standard Lambda-CMD cosmology.

Although the article is behind a paywall (which is somewhat strange in cosmology, but I digress), you can check other articles by the same author that also use the "varying constants" framework, for example https://arxiv.org/abs/2201.11667. His framework is that the speed of light c, the Planck constant h, the Boltzmann constant k and the Gravitational constant G depend directly on time, or to be more precise, on the expansion factor of the universe. There are two big differences with respect to what you were saying:

• c, h and k are not coupling constants, and therefore they don't receive any energy-dependent corrections. In fact, you could think of these constants as "conversion factors" between units: c converts space-time coordinates in seconds to space-time coordinates in meters, k converts kinetic energy in Joules (or electronvolts) to kinetic energy in Kelvin, and h converts angular momentum or action measured in quanta to angular momentum or action measured in J·s (or eV·s). Honestly it doesn't make much sense to me that these constants could change (what does it means, in physical terms), that they could change in a correlated way, or that they could change in a correlated way to one, and only one, coupling constant, G.
• Since this is a time-dependent change, there is no real way to significantly test the hypothesis (unlike the energy-dependent changes). We can not go back in time, or to wait to a different time when these constants would be different. He actually proposes to study how the experimental determinations of these constants in the last 10-20 years, which sounds very wild, as those tiny differences are very susceptible, by definition, to experimental uncertainties, and they are not very suitable for controlled tests.

It didn’t as time is relative just like space. There is no absolute standard of time to say “time moves faster”. Faster relative to what?