  This work continues the investigation in two recent papers on the quantum
thermodynamics of spacetimes, 1) placing what was studied in [1] for thermal
quantum fields in the context of early universe cosmology, and 2) extending the
considerations of vacuum compressibility of dynamical spaces treated in [2] to
dynamical spacetimes with thermal quantum fields. We begin with a warning that
thermal equilibrium condition is not guaranteed to exist or maintained in a
dynamical setting and thus finite temperature quantum field theory in
cosmological spacetimes needs more careful considerations than what is often
described in textbooks. A full description requires nonequilibrium quantum
field theory in dynamical spacetimes using `in-in' techniques. A more
manageable subclass of dynamics is where thermal equilibrium conditions are
established at both the beginning and the end of evolution are both well
defined. Here we shall assume an in-vacuum state. It has been shown that if the
intervening dynamics has an initial period of exponential expansion, such as in
inflationary cosmology, particles created from the parametric amplification of
the vacuum fluctuations in the initial vacuum will have a thermal spectrum
measured at the out-state. Under these conditions finite temperature field
theory can be applied to calculate the quantum thermodynamic quantities. Here
we consider a massive conformal scalar field in a closed four-dimensional
Friedmann-Lemaitre-Robertson-Walker universe based on the simple analytically
solvable Bernard-Duncan model. We calculate the energy density of particles
created from an in-vacuum and derive the partition function. From the free
energy we then derive the heat capacity and the quantum compressibility of the
spacetimes with thermal particle creation. We end with some discussions and
suggestions for further work in this program of studies.
