  In this work we examine the breathing mode of a strongly interacting
two-dimensional Fermi gas and the role of temperature on the anomalous breaking
of scale invariance. By calculating the equation of state with different
many-body $T$-matrix theories and the virial expansion approach, we obtain a
hydrodynamic equation of the harmonically trapped Fermi gas (with trapping
frequency $\omega_{0}$) through the local density approximation. By solving the
hydrodynamic equation we determine the breathing mode frequencies as functions
of interaction strength and temperature. We find that the breathing mode
anomaly depends sensitively on both interaction strength and temperature. In
particular, in the strongly interacting regime we predict a significant
down-shift of the breathing mode frequency, below the scale invariant value of
$2\omega_{0}$ for temperatures of order the Fermi temperature.
