  Temperature uncertainty of a quantum system in canonical ensemble is
inversely determined by its energy fluctuation, which is known as the
temperature-energy uncertainty relation. No such uncertainty relation was
discovered for a non-equilibrium open quantum system. In this article, we
derive a universal temperature uncertainty relation for general non-equilibrium
processes. We find that it is the fluctuation of heat, which is defined as the
change in bath energy, determines the temperature uncertainty in
non-equilibrium case. Specifically, the heat is divided into trajectory heat
and backaction heat, which are associated with the system's trajectory of
evolution and the backaction of partial measurement on system, respectively.
Based on this decomposition, we reveal that both correlations between system
and bath's process function and state function are the resources for enhancing
temperature precision. Our findings are conductive to design ultrahigh
sensitive quantum thermometer.
