  Unimodular gravity is based on a modification of the usual Einstein-Hilbert
action that allows one to recover general relativity with a dynamical
cosmological constant. It also has the interesting property of providing, as
the momentum conjugate to the cosmological constant, an emergent clock
variable. In this paper we investigate the cosmological reduction of unimodular
gravity, and its quantization within the framework of flat homogeneous and
isotropic loop quantum cosmology. It is shown that the unimodular clock can be
used to construct the physical state space, and that the fundamental features
of the previous models featuring scalar field clocks are reproduced. In
particular, the classical singularity is replaced by a quantum bounce, which
takes place in the same condition as obtained previously. We also find that
requirement of semi-classicality demands the expectation value of the
cosmological constant to be small (in Planck units). The relation to spin foam
models is also studied, and we show that the use of the unimodular time
variable leads to a unique vertex expansion.
