  We discuss heat transport in a thermally-biased SQUID in the presence of an
external magnetic flux, when a non-negligible inductance of the SQUID ring is
taken into account. A properly sweeping driving flux causes the thermal current
to modulate and behave hysteretically. The response of this device is analysed
as a function of both the hysteresis parameter and degree of asymmetry of the
SQUID, highlighting the parameter range over which hysteretic behavior is
observable. Markedly, also the temperature of the SQUID shows hysteretic
evolution, with sharp transitions characterized by temperature jumps up to,
e.g., ~ 38mK for a realistic Al-based setup. In view of these results, the
proposed device can effectively find application as a temperature-based
superconducting memory element, working even at GHz frequencies by suitably
choosing the superconductor on which the device is based.
