  We consider a dilute granular gas of hard spheres colliding inelastically
with coefficients of normal and tangential restitution $\alpha$ and $\beta$,
respectively. The basic quantities characterizing the distribution function
$f(\mathbf{v},\bm{\omega})$ of linear ($\mathbf{v}$) and angular
($\bm{\omega}$) velocities are the second-degree moments defining the
translational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. The
deviation of $f$ from the Maxwellian distribution parameterized by
$T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associated
with the fourth-degree velocity moments. The main objective of this paper is
the evaluation of the collisional rates of change of these second- and
fourth-degree moments by means of a Sonine approximation. The results are
subsequently applied to the computation of the temperature ratio
$T^\text{rot}/T^\text{tr}$ and the cumulants of two paradigmatic states: the
homogeneous cooling state and the homogeneous steady state driven by a
white-noise stochastic thermostat. It is found in both cases that the
Maxwellian approximation for the temperature ratio does not deviate much from
the Sonine prediction. On the other hand, non-Maxwellian properties measured by
the cumulants cannot be ignored, especially in the homogeneous cooling state
for medium and small roughness. In that state, moreover, the cumulant directly
related to the translational velocity differs in the quasi-smooth limit
$\beta\to -1$ from that of pure smooth spheres ($\beta=-1$). This singular
behavior is directly related to the unsteady character of the homogeneous
cooling state and thus it is absent in the stochastic thermostat case.
