Dancing David
Penultimate Amazing
Where there is a lot of mass light will bend and time will slow down.
If at the other end of the time spectrum, for light to remain the same speed in all frames, then it must speed up.
Having light speed up is like stepping on one of physics' landmines, you really don't want to do that. I was hoping not to step on that mine. Looks like I might need some earplugs.
No, that is part of GR, when light's path changes, it is because the space curvature is changed. Time is a related metric but you need to stop pulling it out.
Light does not speed up, time dialation occurs because of the invariant speed of light, I can't say it more plainly.