Notice first that the phase space of any theory is nothing but the space of all its classical solutions. The traditional presentation of phase spaces by fields and their canonical momenta on a Cauchy surface is just a way of parameterizing all solutions by initial value data -- if possible. This is often possible, but comes with all the disadvantages that a choice of coordinates always comes with. The phase space itself exists independently of these choices and whether they exist in the first place. In order to emphasize this point one sometimes speaks of covariant phase space .
This is well known, even if it remains a bit hidden in many textbooks. For more details and an extensive and commented list of references on this see the $n$Lab entry phase space .
Then notice that the phase space of every field theory that comes from a local action functional (meaning that it is the integral of a Lagrangian which depends only on finitely many derivatives of the fields) comes canonically equipped with a canonical Liouville form and a canonical presymplectic form. The way this works is also discuss in detail at phase space . A good classical reference is Zuckerman, a more leisurely discussion is in Crncovic-Witten .
This canonical presymplectic form that exists on the phase space of every local theory becomes symplectic on the reduced phase space, which is the space obtained by quotienting out the gauge symmetries. This quotient is often very ill-behaved, but it always exists nicely as a "derived" quotient, and as such is modeled by the BV-BRST complex (as discussed there). The whole (Lagrangian) BV-BRST machinery is there to produce the canonical symplectic form existing on the reduced phase space of any local action functional.
Since the Einstein-Hilbert action and all of its usual variants with matter couplings etc. is a local action functional, all this applies to gravity. Recently Fredenhagen et al. have given careful discussions of the covariant phase space of gravity (and its Liouville form), see the references listed here .
It follows that the "dimension" of the covariant phase space of gravity does not depend on the "size of the universe", nor does it make much sense to ask this, in the first place. A given cosmology is one single point in this phase space (or rather it is so in the reduced phase space, after quotienting out symmetries).
However, you might be after some truncations or effective approximations or coarse graining to full covariant gravity. For these the story might be different.