Q:

Does General Relativity bring "Cosmic Time," which is the same for all frames-of-reference?

In a recent podcast, the philosopher William Lane Craig said that while Special Relativity taught us to doubt whether any given clock is, or whether it could be, an absolute measure of time, General Relativity restored the possibility and perhaps evidence for a single "parameter": "Cosmic Time", which is the same for all frames-of-reference.

[...] what I point out in my published work is that when you turn to General Relativity then Absolute Time reemerges on a cosmic scale
  that was denied in Special Theory. So the Special Theory has been superseded by
  General Theory of Relativity. In General Relativity, there emerges a cosmic time, Kevin, that measures the duration of the universe from its inception. And this cosmic time is the same for every observer in the universe, regardless
  of his state of motion. It is not relative to reference frames or motion. It is frame independent (and in that sense absolute) and it measures the absolute duration of the universe.

Is this true?  Have physicists found in General Relativity some evidence that there is in fact a single time parameter that both of the twins in the twins paradox, together with us and the most distant astronomical object, on our side and the distant side of the cosmological horizon... everything "knows" the true age of the Universe?

A:

No, cosmic time is only universal for observers moving just with the expansion of the universe.  
As explained in M. Pettini: Introduction to Cosmology:  

An observer at rest with respect to the substratum is a fundamental observer. If the substratum is in motion, we say that the class of fundamental
  observers are comoving with the substratum. (italics in original)  

...  

An immediate consequence of homogeneity is the existence of a universal
  cosmic time, t. Since all fundamental observers see the same sequence of
  events in the universe, they can synchronise their clocks by means of these
  events. (italics in original)

So the universe on a large scale is moving like a fluid,and more particularly in our actual universe it is expanding. Observers moving only with the expansion are fundamental observers experiencing cosmic time.  

