We consider gravitational path-integral of JT gravity of closed universe in AdS and dS spacetimes. In the absence of topology change, we calculate the amplitude and identify the perturbative Hilbert space. We find the Hilbert space is given by the co-invariant Hilbert space, an extended Hilbert space of boundary conditions divided by constraints, which is infinite dimensional. We also propose a finite N regularization of the model, whose Hilbert space dimension can be non-trivial