  The dimension of the Hilbert space of a quantum gravitational system can be
written formally as a path integral partition function over Lorentzian metrics.
We implement this in a 2+1 dimensional simplicial minisuperspace model in which
the system is a spatial topological disc, and recover by contour deformation
through a Euclidean saddle the entropy of the de Sitter static patch, up to
discretization artifacts. The model illustrates the importance of integration
over both positive and negative lapse to enforce the gravitational constraints,
and of restriction to complex metrics for which the fluctuation integrals would
converge. Although a strictly Lorentzian path integral is oscillatory, an
exponentially large partition function results from unavoidable imaginary
contributions to the action. These arise from analytic continuation of the
simplicial (Regge) action for configurations with codimension-2 simplices where
the metric fails to be Lorentzian. In particular, the dominant contribution
comes from configurations with contractible closed timelike curves that
encircle the boundary of the disc, in close correspondence with recent
continuum results.
