  We theoretically study the thermal relaxation of many-body systems under the
action of oscillating external fields. When the magnitude or the orientation of
a field is modulated around values where the pairwise heat-exchange
conductances depend non-linearly on this field, we demonstrate that the time
symmetry is broken during the evolution of temperatures over a modulation
cycle. We predict that this asymmetry enables a pumping of heat which can be
used to cool down faster the system. This effect is illustrated through
different magneto-optical systems under the action of an oscillating magnetic
field.
