Speed of light in different parts of the universe Do we have enough evidence to claim that speed of light remains the same at 'c' even at the most distant parts of the visible universe? 
 A: Quite simply, we have very little evidence to suggest that we know much of anything about the behavior of light non-locally. Emil Wolf showed that the Mutual Coherence Function for any two electric fields completely defines the behavior in the far field. While this might initally seem to be proof for the validity of assuming that the properties of light remain constant, if there is any perturbation in which two electric fields interact in between our observation point and the source then that perturbation acts as a new source for our observation point. It is the same basic principle as a hologram (although not restricted to coherent light). Although the source is well defined, because there is something in the way, what you see is not the source at all but rather an image formed by the interaction of the electric field.
We claim that the speed of light remains the same everywhere in the universe because we assume that space is predominantly empty. Thus, with nothing to get in the way of the light, it will travel at the speed of light. While this may be true, it may also be true that the universe only appears empty because of the nature of the interactions of the wave field according to the mutual coherence function. However, for the progress of astronomy it is better to assume that our observations carry some direct relation to the reality of what we are observing. Only exploration will prove one way or the other.
A: The claim that signal front speed (a.k.a. speed of light in vacuum, "$c_0$" or for short "$c$") is a universal constant is not dependent on (or consequence of) any evidence but due to definition (foremost within the theory of relativity) how geometric or kinematic relations ought to be measured, and how thereby evidence ought to be gathered, in the first place.
How J.L.Synge put it ("Relativity. The general theory", Ch. III §2):

 For us time [duration] is the only basic measure. Length (or distance), in so far as it is necessary or desirable to introduce it, is strictly a derived concept [and consequently, so is speed].

As long as the symbol letter $c$ is consistently used in deriving values of distances and (subsequently) values of speed, and as long as the same measurement procedures are used to determine duration values (or at least real number values of duration ratios) in the first place, it is rightly claimed that "the speed of light [in vacuum] remains the same " in each trial (anywhere/anytime in the universe) in which geometric or kinematic relations were measured.
Quite a different and more difficult question, however, is how to determine the values of refractive indices, i.e. how some region might differ from having been "(optical) vacuum" with respect to the propagation of electro-magnetic fields (e.g. of given frequency, or frequency range); especially for distant regions of the universe.  
