A new pre-print calculates the mass of the graviton in a massive gravity theory consistent with the single parameter that explains galactic rotation curves in MOND theory, the Hubble constant, the proportion of mass-energy in the universe that is dark energy (Omega lambda), the speed of light, and Planck's constant. The formula is h bar/c^2 times the Hubble constant times the square root of three times Omega lambda.
The result is 4*10^-69 kilograms, which is equivalent to 2*10^-33 eV/c^2, which is equivalent to 2*10^-61 times the Planck mass. The Compton wavelength of a graviton with this mass would be on the order of 100,000 light years (roughly the same as the diameter of the Milky Way galaxy).
Massive gravity theories are candidates for resolving both the dark energy and the dark matter problems of astrophysics.
I am skeptical that gravitons have rest mass. But, experimental evidence cannot rule out of graviton mass of the scale suggested: the current experimental limit is less than 6*10^-32 eV/c^2 (about thirty times heavier than the predicted value in this preprint).
Gravitons, if they exist at all, however, clearly must have mass-energy, however. And, massive gravity theory is one way to tease out what general relativity would look like if it treated the mass-energy of gravitons as a source of gravitation curvature (which Einstein's field equations do not).