  In a recent paper [A. Santos, G. M. Kremer, and V. Garz\'o, \emph{Prog.
Theor. Phys. Suppl.} \textbf{184}, 31-48 (2010)] the collisional energy
production rates associated with the translational and rotational granular
temperatures in a granular fluid mixture of inelastic rough hard spheres have
been derived. In the present paper the energy production rates are explicitly
decomposed into equipartition rates (tending to make all the temperatures
equal) plus genuine cooling rates (reflecting the collisional dissipation of
energy). Next the homogeneous free cooling state of a binary mixture is
analyzed, with special emphasis on the quasi-smooth limit. A previously
reported singular behavior (according to which a vanishingly small amount of
roughness has a finite effect, with respect to the perfectly smooth case, on
the asymptotic long-time translational/translational temperature ratio) is
further elaborated. Moreover, the study of the time evolution of the
temperature ratios shows that this dramatic influence of roughness already
appears in the transient regime for times comparable to the relaxation time of
perfectly smooth spheres.
