In article , "Richard D. Saam"
writes:

Thus the critical density is, by definition, 3H^2/(8\pi G) since the
value of the density is always the critical density in the Einstein-de
Sitter universe. So H(z) is proportional to the square root of the
density. (If lamnda and/or k are not zero, then the expression for H(z)
is of course more complicated.)

In a purely mathematical sense
critical density rho_c is proportional to H^2/G
and
critical density rho_c is proportional to H^2
if G is a constant.
but alternatively
critical density rho_c is proportional to H
if H/G is a constant.

As previously posted by Jonathan Thornburg,
lunar ranging in our local bound system indicates
(d^2G/dt^2)/G = (4 ± 5) * 10^{-15}/year^2
but this does not negate
a delta G in galactic cluster systems
in near equilibrium with expanding critical density rho_c.

There are also constraints on the variability of G over cosmological
distances. Not as strict as the local constraints, but strict enough to
rule out it being important for explaining something---at best, one
could hope to marginally DETECT a variation.