  The homogeneous cooling state (HCS) of a granular gas described by the
inelastic Boltzmann equation is reconsidered. As usual, particles are taken as
inelastic hard disks or spheres, but now the coefficient of normal restitution
$\alpha$ is allowed to take negative values $\alpha\in[-1,1]$, a simple way of
modeling more complicated inelastic interactions. The distribution function of
the HCS is studied at the long-time limit, as well as for intermediate times.
At the long-time limit, the relevant information of the HCS is given by a
scaling distribution function $\phi_s(c)$, where the time dependence occurs
through a dimensionless velocity $c$. For $\alpha\gtrsim -0.75$, $\phi_s$
remains close to the gaussian distribution in the thermal region, its cumulants
and exponential tails being well described by the first Sonine approximation.
On the contrary, for $\alpha\lesssim -0.75$, the distribution function becomes
multimodal, its maxima located at $c\ne 0$, and its observable tails algebraic.
The latter is a consequence of an unbalanced relaxation-dissipation
competition, and is analytically demonstrated for $\alpha\simeq -1$ thanks to a
reduction of the Boltzmann equation to a Fokker-Planck-like equation. Finally,
a generalized scaling solution to the Boltzmann equation is also found
$\phi(c,\beta)$. Apart from the time dependence occurring through the
dimensionless velocity, $\phi(c,\beta)$ depends on time through a new parameter
$\beta$ measuring the departure of the HCS from its long-time limit. It is
shown that $\phi(c,\beta)$ describes the time evolution of the HCS for almost
all times. The relevance of the new scaling is also discussed.
