  We evaluate the quantum gravity partition function that counts the dimension
of the Hilbert space of a simply connected spatial region of fixed proper
volume in the context of Lovelock gravity, generalizing the results for
Einstein gravity [1]. We find that there exists sphere saddle metrics for a
partition function at a fixed spatial volume in Lovelock theory. Those
stationary points take exactly the same forms as in Einstein gravity. The
logarithm of Z corresponding to a zero effective cosmological constant
indicates the Bekenstein-Hawking entropy of the boundary area and the one
corresponding to a positive effective cosmological constant points to the Wald
entropy of the boundary area. We also observe the existence of zeroth order
phase transitions between different vacua, a phenomenon distinct from Einstein
gravity.
