  We study the conditions under which a Mpemba-like effect emerges in granular
gases of inelastic and rough hard disks driven by a class of thermostats
characterized by the splitting of the noise intensity into translational and
rotational counterparts. Thus, granular particles are affected by a stochastic
force and a stochastic torque, which inject translational and rotational
energy, respectively. We realize that a certain choice of a thermostat of this
class can be characterized just by the total intensity and the fraction of
noise transferred to the rotational degree of freedom with respect to the
translational ones. Firstly, Mpemba effect is characterized by the appearance
of a crossing between the temperature curves of the considered samples. Later,
an overshoot of the temperature evolution with respect to the steady-state
value is observed and the mechanism of Mpemba effect generation is changed. The
election of parameters allows to design plausible protocols based on these
thermostats for generating the initial states to observe the Mpemba-like effect
in experiments. In order to obtain explicit results, we use a well-founded
Maxwellian approximation for the evolution dynamics and the steady-state
quantities. Finally, theoretical results are compared with direct simulation
Monte Carlo and molecular dynamics results, and a very good agreement is found.
