General relativity does indeed predict that, in a gravitational field, the speed of light varies twice as fast as the speed of ordinary matter. Yet the (Newtonian) truth is that the two variations are identical - the variation of the speed of light obeys the equation c'=c(1+gh/c^2) given by Newton's emission theory of light and explicitly used by Einstein in 1911. In the final version of general relativity however there is a factor of 2 on the potential term (c'=c(1+2gh/c^2)):
http://www.ita.uni-heidelberg.de/research/bartelmann/Publications/Proceedings/JeruLect.pdf LECTURES ON GRAVITATIONAL LENSING, RAMESH NARAYAN AND MATTHIAS BARTELMANN, p. 3: " The effect of spacetime curvature on the light paths can then be expressed in terms of an effective index of refraction n, which is given by (e.g. Schneider et al. 1992): n = 1-(2/c^2)phi = 1+(2/c^2)|phi| Note that the Newtonian potential is negative if it is defined such that it approaches zero at infinity. As in normal geometrical optics, a refractive index n>1 implies that light travels slower than in free vacuum. Thus, the effective speed of a ray of light in a gravitational field is: v = c/n ~ c-(2/c)|phi| "
The Newtonian equation c'=c(1+gh/c^2) is obviously consistent with the frequency shift f'=f(1+gh/c^2) measured in the Pound-Rebka experiment. The Einsteinian equation c'=c(1+2gh/c^2) is obviously inconsistent with this frequency shift.