  We combine the Macroscopic Fluctuation Theory and the Inverse Scattering
Method to determine the full long-time statistics of the energy density
$u(x,t)$ averaged over a given spatial interval, $$ U
=\frac{1}{2L}\int_{-L}^{L}dx\, u(x,t)\,, $$ in a freely expanding
Kipnis-Marchioro-Presutti (KMP) lattice gas on the line, following the release
at $t=0$ of a finite amount of energy at the origin. In particular, we show
that, as time $t$ goes to infinity at fixed $L$, the large deviation function
of $U$ approaches a universal form when expressed in terms of the energy
content of the interval $|x|<L$ at time $t$. A key part of the solution is the
determination of the most likely configuration of the energy density at time
$t$, conditional on $U$.
