  Conditions for the stability under linear perturbations around the
homogeneous cooling state are studied for dilute granular gases of inelastic
and rough hard disks or spheres with constant coefficients of normal ($\alpha$)
and tangential ($\beta$) restitution. After a formally exact linear stability
analysis of the Navier--Stokes--Fourier hydrodynamic equations in terms of the
translational ($d_t$) and rotational ($d_r$) degrees of freedom, the transport
coefficients derived in the companion paper [A. Meg\'ias and A. Santos,
"Hydrodynamics of granular gases of inelastic and rough hard disks or spheres.
I. Transport coefficients," Phys. Rev. E 104, 034901 (2021)] are employed.
Known results for hard spheres [V. Garz\'o, A. Santos, and G. M. Kremer, Phys.
Rev. E 97, 052901 (2018)] are recovered by setting $d_t=d_r=3$, while novel
results for hard disks ($d_t=2$, $d_r=1$) are obtained. In the latter case, a
high-inelasticity peculiar region in the $(\alpha,\beta)$ parameter space is
found, inside which the critical wave number associated with the longitudinal
modes diverges. Comparison with event-driven molecular dynamics simulations for
dilute systems of hard disks at $\alpha=0.2$ shows that this theoretical region
of absolute instability may be an artifact of the extrapolation to high
inelasticity of the approximations made in the derivation of the transport
coefficients, although it signals a shrinking of the conditions for stability.
In the case of moderate inelasticity ($\alpha=0.7$), however, a good agreement
between the theoretical predictions and the simulation results is found.
