cosmology - Does the dimensionality of phase space go up as the universe expands?

Ever since Hubble, it is well known that the universe is expanding from a Big Bang. The size of the universe had gone up by many many orders of magnitude as space expanded. If the dimensionality of the quantum phase space is finite because of spatial cutoffs at the Planck scale, does it go up as space expanded? If yes, how can this be squared with unitarity? If no, would this lead to what Tegmark called the Big Snap where something has got to give sometime. What is that something which gives?