Solar luminosity proportional to $G^7$?
I think the argument is the following. The central temperature can be estimated from a form of the virial theorem. At least dimensionally speaking,  total thermal energy $MkT/\mu$ (where $\mu$ is the average mass per particle) is proportional to gravitational potential energy $GM^2/R$. So $T \propto GM/R$.

But for a given mass, standard polytropic theory for a star, says that 
$$M \propto \rho_c^{(3-n)/2n} G^{-3/2}$$
$$R  \propto \rho_c^{(1-n)/2n} G^{-1/2}$$
 where $n=3$ is an approximate polytropic index for the Sun and $\rho_c$ is the central density. See for example Does Eddington&#39;s variable polytropic index better fit data from the Standard Solar Model?

Putting this together we have 
$$ T \propto GMR^{-1} \propto G^2 \rho_c^{-1/3}$$

The power generation in the solar core from the pp cycle goes approximately as $L \propto \rho_c^2 T^4$ and using the proportionality for $T$ derived above, we have $L \propto G^8 \rho_c^{2/3}$.

Finally, we note that for a $n=3$ polytrope, that $\rho _c$ and $M$ are independent, but that $\rho_c^{2/3} \propto G^{-1} R^{-2}$. So for a fixed solar radius we have $L \propto G^7$.