  We consider a free particle coupled with finite strength to a bath and
investigate the evaluation of its specific heat. A harmonic oscillator bath of
Drude type with cutoff frequency omega_D is employed to model an ohmic friction
force with dissipation strength gamma. Two scenarios for obtaining specific
heat are presented. The first one uses the measurement of the kinetic energy of
the free particle while the second one is based on the reduced partition
function. Both descriptions yield results which are consistent with the Third
Law of thermodynamics. Nevertheless, the two methods produce different results
that disagree even in their leading quantum corrections at high temperatures.
We also consider the regime where the cutoff frequency is smaller than the
friction strength, i.e. omega_D<gamma. There, we encounter puzzling results at
low temperatures where the specific heat based on the thermodynamic
prescription becomes negative. This anomaly is rooted in an ill-defined density
of states of the damped free particle which assumes unphysical negative values
when gamma/omega_D>1.
