  We analyze the full-counting statistics of the electric heat current flowing
in a two-terminal quantum conductor whose temperature is probed by a third
electrode ("probe electrode"). In particular we demonstrate that the
cumulant-generating function obeys the fluctuation theorem in the presence of a
constant magnetic field. The analysis is based on the scattering matrix of the
three-terminal junction (comprising of the two electronic terminals and the
probe electrode), and a separation of time scales: it is assumed that the rapid
charge transfer across the conductor and the rapid relaxation of the electrons
inside the probe electrode give rise to much slower energy fluctuations in the
latter. This separation allows for a stochastic treatment of the probe
dynamics, and the reduction of the three-terminal setup to an effective
two-terminal one. Expressions for the lowest nonlinear transport coefficients,
e.g., the linear-response heat-current noise and the second nonlinear thermal
conductance, are obtained and explicitly shown to preserve the symmetry of the
fluctuation theorem for the two-terminal conductor. The derivation of our
expressions which is based on the transport coefficients of the three-terminal
system explicitly satisfying the fluctuation theorem, requires the full
calculations of vertex corrections.
