What is the experimental evidence that the gravitational constant $G$ runs with energy (or temperature)? Various papers on quantum field theory claim that $G$ runs with energy, like the fine structure constant does. Some examples:
Frolov, Fursaev and Zelnikov, Nucl.Phys. B486, 339 (1997)
Volovik  and  Zelnikov,  JETP  Lett. 78,  751 (2003)
Hamber  and  Williams,  Phys.  Rev.  D75, 084014 (2007)
(Some people call this running with temperature.)
What is the experimental situation? Is there any evidence for such a running? Or against it?
 A: There is no direct experimental evidence that $G$ runs with energy.
There is direct experimental evidence that other constants in nature run with energy due to quantum effects, such as the fine structure constant (see for instance https://en.wikipedia.org/wiki/Fine-structure_constant#Variation_with_energy_scale). This running is a well-understood consequence of the renormalization group and can be calculated within the quantum field theory of the Standard Model, and the predictions agree with experiment.
Applying the same logic to gravity, many theorists expect that $G$ will run with energy and can even calculate how it will run, under some assumptions about gravity acting like other quantum field theories (at least at energies below the Planck scale), and what matter fields exist in the Universe. These calculations are interesting to a certain class of mathematically minded theorists, but the sad numerical reality is that the running is utterly irrelevant for any system we can probe experimentally.
