The dipole anisotropy of the Cosmic Microwave Background will continue to be a good tool to see the constancy of the speed of light for all observers into the practically infinite future, assuming the infinite future looks a lot like de Sitter space (which is essentially what the standard cosmology predicts).

http://www.astro.ucla.edu/~wright/CMB-dipole-history.html

The far future of the universe will still have generally covariant physics, as does its far past, sharpening up your second sentence.