  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.
