What are the concrete technical arguments supporting the idea that the wave function of the universe can be written as partition function? In this talk, the not yet settled down idea that the wave function of the universe could potentially be written as  the partition function of a scale invariant statistical field theory is mentioned:

$$
\Psi[g] = Z[g] = \int D(\text{Fields}) e^{-S[g,\text{Fields}]}
$$

If our universe were AdS, this relation could already be well enough explained by the AdS/CFT correspondence, but as our expanding universe correspond to a dS geometry, things are less clear.

What are the concrete technical arguments, ideas, or hints that this relationship should hold for our dS universe too? What work has already been done on this?
Similar approaches have been generalized to more general spacetimes.

Technical details for dS space and further references in this paper about the holographic principle, p.43 ff., may provide some idea.