FAQ: Is the speed of light equal to c even in an accelerating frame of reference?
The short answer is "yes."
The long answer is that it depends on what you mean by measuring the speed of light.
In the SI, the speed of light has a defined value of 299,792,458 m/s, because the meter is defined in terms of the speed of light. In the system of units commonly used by relativists, it has a defined value of 1. Obviously we can't do an experiment that will remeasure 1 to greater precision. However, it could turn out to have been a bad idea to give the speed of light a defined value. For example, it would have been a bad idea to give the speed of sound a defined value, because the speed of sound depends on extraneous variables such as temperature.
One such extraneous variable might be the direction in which the light travels, as in the Sagnac effect, which was first observed experimentally in 1913. In the Sagnac effect, a beam of light is split, and the partial beams are sent clockwise and counterclockwise around an interferometer. If the interferometer is rotating in the plane of the beams' path, then a shift is observed in their interference, revealing that the time it takes light to go around the apparatus clockwise is different from the time it takes to go around counterclockwise. An observer in a nonrotating frame explains the observation by saying that the beams went at equal speeds, but their times of flight were unequal because while they were in flight, the apparatus accelerated. An observer in the frame rotating along with the apparatus says that clearly the beams could not have always had the same speed c, since they took unequal times to travel the same path. If we insist on letting c have a defined value, then the rotating observer is forced to say that the same closed path has a different length depending on whether the length is measured clockwise or counterclockwise. This is equivalent to saying that the distance unit has a length that depends on whether length is measured clockwise or counterclockwise.
Silly conclusions like this one can be eliminated by specifying that c has a defined value not in all experiments but in local experiments. The Sagnac effect is nonlocal because the apparatus has a finite size. The observed effect is proportional to the area enclosed by the beam-path. "Local" is actually very difficult to define rigorously [Sotiriou 2007], but basically the idea is that if your apparatus is of size L, any discrepancy in its measurement of c will approach zero in the limit as L approaches zero.
In a curved spacetime, it is theoretically possible for electromagnetic waves in a vacuum to undergo phenomena like refraction and partial reflection. Such effects are far too weak to be detected by any foreseeable technology. Assuming that they do really exist, they could be seen as analogous to what one sees in a dispersive medium. The question is then whether this constitutes a local effect or a nonlocal one. Only if it's a local effect would it violate the equivalence principle. This is closely related to the famous question of whether falling electric charges violate the equivalence principle. The best known paper on this is DeWitt and DeWitt (1964). A treatment that's easier to access online is Gron and Naess (2008). You can find many, many papers on this topic going back over the decades, with roughly half saying that such effects are local and violate the e.p., and half saying they're nonlocal and don't.
Sotiriou, Faraoni, and Liberati, arxiv.org/abs/0707.2748
Cecile and Bryce DeWitt, "Falling Charges," Physics 1 (1964) 3
Gron and Naess, arxiv.org/abs/0806.0464v1