  We consider the quantum gravity partition function that counts the dimension
of the Hilbert space of a spatial region with topology of a ball and fixed
proper volume, and evaluate it in the leading order saddle point approximation.
The result is the exponential of the Bekenstein-Hawking entropy associated with
the area of the saddle ball boundary, and is reliable within effective field
theory provided the mild curvature singularity at the ball boundary is
regulated by higher curvature terms. This generalizes the classic
Gibbons-Hawking computation of the de Sitter entropy for the case of positive
cosmological constant and unconstrained volume, and hence exhibits the
holographic nature of nonperturbative quantum gravity in generic finite volumes
of space.
