Time-translation symmetry breaking is a mechanism for the emergence of
non-stationary many-body phases, so-called time-crystals, in Markovian open
quantum systems. Dynamical aspects of time-crystals have been extensively
explored over the recent years. However, much less is known about their
thermodynamic properties, also due to the intrinsic nonequilibrium nature of
these phases. Here, we consider the paradigmatic boundary time-crystal system,
in a finite-temperature environment, and demonstrate the persistence of the
time-crystalline phase at any temperature. Furthermore, we analyze
thermodynamic aspects of the model investigating, in particular, heat currents,
power exchange and irreversible entropy production. Our work sheds light on the
thermodynamic cost of sustaining nonequilibrium time-crystalline phases and
provides a framework for characterizing time-crystals as possible resources
for, e.g., quantum sensing. Our results may be verified in experiments, for
example with trapped ions or superconducting circuits, since we connect
thermodynamic quantities with mean value and covariance of collective
(magnetization) operators.Comment: 7+6 pages, 2+1 figure