  We study the thermodynamic properties of a system of two-level dipoles that
are coupled ultrastrongly to a single cavity mode. By using exact numerical and
approximate analytical methods, we evaluate the free energy of this system at
arbitrary interaction strengths and discuss strong-coupling modifications of
derivative quantities such as the specific heat or the electric susceptibility.
From this analysis we identify the lowest-order cavity-induced corrections to
those quantities in the collective ultrastrong coupling regime and show that
for even stronger interactions the presence of a single cavity mode can
strongly modify extensive thermodynamic quantities of a large ensemble of
dipoles. In this non-perturbative coupling regime we also observe a significant
shift of the ferroelectric phase transition temperature and a characteristic
broadening and collapse of the black-body spectrum of the cavity mode. Apart
from a purely fundamental interest, these general insights will be important
for identifying potential applications of ultrastrong-coupling effects, for
example, in the field of quantum chemistry or for realizing quantum thermal
machines.
