I know that there is a similar questions, but I think mine is a bit different. I wonder if with the expansion of the universe the speed of light changes. It seems that the speed of light is very connected to the space-time continuum, e.g. it follows all it's deformations etc. So if the space-time continuum expands, does the speed of light change?
In general relativity we describe the geometry of spacetime using a function called the metric. This metric is derived from the distribution of matter - indeed solving Einstein's equations normally involves starting with some distribution of matter and solving the equations to calculate the metric.
If we assume that matter is approximately spread out evenly in the universe then we end up with a surprisingly simple form for the metric. This is called the The Friedmann–Lemaître–Robertson–Walker metric and it is this metric that describes the expanding universe. When a relativist talks about the expanding universe they almost invariably mean the FLRW metric (or something closely related).
The FLRW metric is:
$$ c^2d\tau^2 = c^2dt^2 - a^2(t)(dx^2 + dy^2 + dz^2) $$
Exactly what the equation means is somewhat opaque to non-GR nerds, but the relevant fact here is that the parameter $c$ is a constant. With some jiggery-pokery we can work out what the constant $c$ means, and it turns out to be the speed of light.
So in general relativity the speed of light (technically only the local speed of light) is a constant because GR assumes it's constant. This is an important point to appreciate - GR doesn't prove $c$ is a constant it assumes $c$ is a constant. So the answer to your question is that whether the universe is expanding, contracting or turning cartwheels the speed of light is always constant.
GR could be wrong, and one day an experiment could demonstrate that the speed of light isn't constant. The trouble is that GR is pretty well supported by experiments, and of course the recent detection of gravitational waves adds more support. There are competing theories in which the speed of light can be variable but (so far) none them have proven a better description of the universe than general relativity.
Well, an axiom that is "measured and determined" is not really an axiom :-)
If the speed of light was different in another time or place in the visible universe, we could see it at least indirectly, since c occurs in a lot of physical balances, e.g., the structure or fine structure of bands in the spectrum of light emission or absorption by stars and gazs.
And for a start, the condition and result of creation of matter, molecules, galaxies, stars, would also be different.
We don't observe that, so it's reasonnable to say that in all the visible universe, and in the past since the formation of the first galaxies, c has not changed.