Some say it does. I was taught, that anything following the geodesics of a curved spacetime accelerates. On the other hand you can argue a technical problem, concerning the constancy of light.
I'm guessing your thinking that acceleration means the speed of light must change, contrary to relativity. Just because something accelerates doesn't mean that its speed changes. A good example is an object in uniform circular motion. It is constantly accelerating, but its speed is constant. Only its direction changes.
