  We investigate path integral quantization of two versions of unimodular
gravity. First a fully diffeomorphism-invariant theory is analyzed, which does
not include a unimodular condition on the metric, while still being equivalent
to other unimodular gravity theories at the classical level. The path integral
has the same form as in general relativity (GR), except that the cosmological
constant is an unspecified value of a variable, and it thus is unrelated to any
coupling constant. When the state of the universe is a superposition of vacuum
states, the path integral is extended to include an integral over the
cosmological constant. Second, we analyze the standard unimodular theory of
gravity, where the metric determinant is fixed by a constraint. Its path
integral differs from the one of GR in two ways: the metric of spacetime
satisfies the unimodular condition only in average over space, and both the
Hamiltonian constraint and the associated gauge condition have zero average
over space. Finally, the canonical relation between the given unimodular
theories of gravity is established.
