Comment on "Background thermal contributions in testing the Unruh effect"

Costa and Matsas (gr-qc/9412030) claimed in their recent paper that a thermal bath does not affect substantially the transition probability for fast moving inertial Unruh detector. It is shown that their claim holds only for small $\beta \Delta E$. We show that, for large enough $\beta \Delta E$, the transition probability is not monotonically decreasing function of the detector's speed contrary to their claim.