  The velocity distribution of a classical gas of atoms in thermal equilibrium
is the normal Maxwell distribution. It is well known that for sub-recoiled
laser cooled atoms L\'evy statistics and deviations from usual ergodic
behaviour come into play. Here we show how tools from infinite ergodic theory
describe the cool gas. Specifically, we derive the scaling function and the
infinite invariant density of a stochastic model for the momentum of laser
cooled atoms using two approaches. The first is a direct analysis of the master
equation and the second following the analysis of Bertin and Bardou using the
lifetime dynamics. The two methods are shown to be identical, but yield
different insights into the problem. In the main part of the paper we focus on
the case where the laser trapping is strong, namely the rate of escape from the
velocity trap is $R(v) \propto |v|^\alpha$ for $v \to 0$ and $\alpha>1$. We
construct a machinery to investigate the time averages of physical observables
and their relation to ensemble averages. The time averages are given in terms
of functionals of the individual stochastic paths, and here we use a
generalisation of L\'evy walks to investigate the ergodic properties of the
system. Exploring the energy of the system, we show that when $\alpha=3$ it
exhibits a transition between phases where it is either an integrable or non
integrable observable, with respect to the infinite invariant measure. This
transition corresponds to very different properties of the mean energy, and to
a discontinuous behaviour of the fluctuations. Since previous experimental work
showed that both $\alpha=2$ and $\alpha=4$ are attainable we believe that both
phases could be explored also experimentally.
