  Finding the proper entropy-like Lyapunov functional associated with the
inelastic Boltzmann equation for an isolated freely cooling granular gas is a
still unsolved challenge. The original $H$-theorem hypotheses do not fit here
and the $H$-functional presents some additional measure problems that are
solved by the Kullback--Leibler divergence (KLD) of a reference velocity
distribution function from the actual distribution. The right choice of the
reference distribution in the KLD is crucial for the latter to qualify or not
as a Lyapunov functional, the asymptotic "homogeneous cooling state" (HCS)
distribution being a potential candidate. Due to the lack of a formal proof far
from the quasielastic limit, the aim of this work is to support this conjecture
aided by molecular dynamics simulations of inelastic hard disks and spheres in
a wide range of values for the coefficient of restitution ($\alpha$) and for
different initial conditions. Our results reject the Maxwellian distribution as
a possible reference, whereas they reinforce the HCS one. Moreover, the KLD is
used to measure the amount of information lost on using the former rather than
the latter, revealing a non-monotonic dependence with $\alpha$.
