The second is defined as the the time taken for a particular 
number of vibrations of the electromagnetic wave associated 
with a particular electronic transition of a particular atom.
The speed of light is defined as distance light travels in one second 
measured as the number of wavelengths of the electromagnetic wave
associated with a particular transition of a particular atom.
If the same transition of the same atom is used for both definitions, 
then the velocity of light will indeed be constant, 
and will be unaffected by time, space and physical conditions, 
not even by warps in space/time. 
This would seem to be a very desirable state of affairs.