  A coarse graining technique akin to block spin transformations that groups
together fiducial cells in a homogeneous and isotropic universe has been
recently developed in the context of loop quantum cosmology. The key technical
ingredient was an SU(1, 1) group and Lie algebra structure of the physical
observables as well as the use of Perelomov coherent states for SU(1, 1). It
was shown that the coarse graining operation is completely captured by changing
group representations. Based on this result, it was subsequently shown that one
can extract an explicit renormalisation group flow of the loop quantum
cosmology Hamiltonian operator in a simple model with dust-clock. In this
paper, we continue this line of investigation and derive a coherent state path
integral formulation of this quantum theory and extract an explicit expression
for the renormalisation-scale dependent classical Hamiltonian entering the path
integral for a coarse grained description at that scale. We find corrections to
the non-renormalised Hamiltonian that are qualitatively similar to those
previously investigated via canonical quantisation. In particular, they are
again most sensitive to small quantum numbers, showing that the large quantum
number (spin) description captured by so called "effective equations" in loop
quantum cosmology does not reproduce the physics of many small quantum numbers
(spins). Our results have direct impact on path integral quantisation in loop
quantum gravity, showing that the usually taken large spin limit should be
expected not to capture (without renormalisation, as mostly done) the physics
of many small spins that is usually assumed to be present in physically
reasonable quantum states.
