Is it theoretically possible that the $G$ constant in Einstein's equation be a functional of all fields present in a given spacetime ? \begin{equation}\tag{1} G_{\mu \nu} + \Lambda \, g_{\mu \nu} = -\, \frac{8 \pi}{c^4} \, G[g, \phi] \, T_{\mu \nu}. \end{equation} Since it's by hypothesis a global functional of some sort defined over all of spacetime, it's still a constant independant of position (so the Bianchi identity and local conservation of energy-momentum still apply), but $G$ may change for different spacetimes and fields configurations. In a sense, it's "scale" dependant.
In other words : Given an asymptotically flat spacetime with matter and total energy $E$ (ADM or Tolman mass), could $G$ be dependant on energy : $G(E)$ ?
Are there any published studies about this idea ?
And as a generalisation, what about the other "constants" of nature ?
Could the cosmological constant $\Lambda$ and the fine structure constant $\alpha \equiv k \, e^2 / \hbar c$ also be some functionals of fields over spacetime ?
What would be the arguments against this idea ?
Note : I'm not asking about position dependance, which isn't the same thing at all : $G = G[g, \phi] \ne G(x)$ (notice the square brackets!). I'm talking about a functional, like the fields action : $S \equiv S[g, \phi]$. So maybe $G$ is proportional to $S$, or any other functional.
EDIT : May the dark matter be explained by such an hypothesis ? If $G$ depends on the scale (i.e the energy involved), then gravity isn't responding the same at our solar system's scale, and at a galactic scale.
This idea has some machian feel, in a way, since properties of curved spacetime may depend on the amount of matter/energy in it, from its $G(E)$ !
Since some people appear to have difficulties with the idea of a functional (not a function), I'm giving a naive example of what $G$ may look like, according to the idea above. For some real scalar field $\phi(x) \sim \mathrm{L}^{-1}$ : \begin{equation}\tag{2} G[g, \phi] = G_0 \, \sqrt{ 1 + G_0^2 \int_{\mathcal{M}} \big( g^{\mu \nu} (\partial_{\mu} \, \phi)(\partial_{\nu} \, \phi) \big)^{2} \, \sqrt{- g} \: d^4 x + \ldots }, \end{equation} where $G_0 \sim \mathrm{L}^2$ is a "naked" gravitationnal constant. The other terms are "scale dependant corrections". So the real gravitational coupling constant $G$ depends globaly on the matter content in the whole of spacetime, or on the scale we consider to do the calculations.
EDIT 2 : Here's a small argument in favor of the previous idea. Ever noticed that both $G$ and $\alpha$ have physical dimensions (i.e units) that depend on the spacetime dimensions $D$ ? (this is well known. Just examine the Poisson equation : $\nabla^2 \phi = 4 \pi G \rho$, where the density $\rho$ depends on the $D - 1$ space dimensions) : \begin{align}\tag{3} G &\sim \mathrm{L}^{D - 2}, &\alpha &\sim \mathrm{L}^{D - 4}. \end{align} Then if their value necessarily changes with the dimensionality $D$ of spacetime, why should they stay the same for all spacetimes of a given dimensionality ?
