According to relativity theory (gravitational time dilation), the observed pace of a clock depends on the strength of the gravitational field at the clock and at the observer. Isn't this at odds with the concept of cosmic time in big bang cosmology: the idea that time passes at the same pace everywhere since if it doesn’t, it makes no sense to speak about the age of the universe?
In cosmology it is assumed that the universe is homogenous and isotropic for all intents and purposes.
Homogeneity means that every point in spacetime is equivalent regarding the dynamics of spacetime. Therefore we can define an observer that has a special physical meaning: the fundamental observer. It is the observer that is stationary with respect to the cosmic fluid.
Cosmic time is defined as that such an observer measured since the big bang. Hence there is no problem with the relativity of time, we have simply defined a time of a specific observer, that is a useful measure in a cosmological context.
The gravitational time dilation and the speed-related time dilation - together with their respective twin paradox - are applying in the whole universe. That means that the age of a photon emitted at the beginning of the universe is zero, and the age of fast-moving particles is very different from what we call the "age of the universe".
You may imagine the universe as a "string curtain" of discrete strings (nothing to do with string theory): each particle is following its own worldline through time, and each particle has its own age, function of its velocity and its exposition to gravitation.
Between particles, space is timeless, because you cannot assign any velocity to points in space as long as there is no particle (at the difference to massless photons to which you can at least assign a proper time zero). This is why foliation of spacetime does not work. All you will get are sections of the "string curtain", but no continuous sheets.
However, there is a solution for the determination of the age of the universe: The age of the slow (non-relativistic) particles is the oldest age which can exist. Slow particles are approximately comoving with the cosmic microwave background, and the resulting age is higher than the one of fast moving particles. That means, the possible error due to relativistic effects with regard to the age of the universe is small.
There may be one reserve which is the early time after the big bang where all particles (even those which stars and planets are made of today) were fast and relativistic. But even in this period we may suppose that there were no particles which were much slower than the particles we are made of, and thus no particles with a longer proper time than the particles stars and planets are made of today.
Laymans answer is coming.
The very simple answer: There is no absolute time in General Relativity.
(There isn't one already in the Special Relativity on a different reason).
The reason in the case of the GR (on my layman's understand) is the following: GR (and already the SR) plays not with time and points in space, it calculates distance and elapsed time between events, i.e. between points of the spacetime. This "spacetime distance" is calculated by an integration on the curved spacetime. But in the general case of the GR, this integration would be ambiguous.
But, the expanding Universe is described by the Friedmann equations. They describe essentially the curvature of the expanding Universe. This curvature is very symmetric, its space-like part is probably flat, so this is a special case, where this integration can be done. This is why we can have a "cosmic time", i.e. we can assign a "time since the big bang" to nearly every spacetime point of the Universe.
In very special situations, for example, around merging black holes, this curveture can be again not so "beautiful", I suspect the "cosmic time" is in their case again impossible.