If I take a spaceship and park it near the event horizon of a black hole and then measure the age of the universe by observing SNe Ia, then travel back out to normal space (no gravitational forces, at rest with respect to CMB), will the dates agree? That is, if the measured age of the universe is 13.8 billion years near the event horizon, and it takes me 100 million years (proper time) to travel back out to normal space, will my new measurement of SNe Ia agree with a date of 13.8 + 0.1 = 13.9 billion years? If that is true, can we say that time is absolute (i.e. all observers will agree on the age of the universe when using SNe Ia when coordinate systems are normalized)?
Well, no. We can construct a much simpler example to see this: fix a point in Minkowski spacetime, and consider two observers following worldlines from that point with a relative velocity. They can even both be inertial. At fixed Minkowski time the two observers measure different proper time.
The FLRW universe, however, is sort of special in that there is a sort of preferred reference frame: that in which the coordinates follow the expansion. This frame is "preferred" in the sense that geodesics which have some relative velocity compared to the expansion asymptote towards it. But you're still allowed to view the solution from an arbitrary coordinate system (for example by considering non-geodesic observers), so you can come up with any "age of the Universe" you like.