  We consider the low-temperature transport properties of critical
one-dimensional systems which can be described, at equilibrium, by a Luttinger
liquid. We focus on the prototypical setting where two semi-infinite chains are
prepared in two thermal states at small but different temperatures and suddenly
joined together. At large distances $x$ and times $t$, conformal field theory
characterizes the energy transport in terms of a single light cone spreading at
the sound velocity $v$. Energy density and current take different constant
values inside the light cone, on its left, and on its right, resulting in a
three-step form of the corresponding profiles as a function of $\zeta=x/t$.
Here, using a non-linear Luttinger liquid description, we show that for generic
observables this picture is spoiled as soon as a non-linearity in the spectrum
is present. In correspondence of the transition points $x/t=\pm v$ a novel
universal region emerges at infinite times, whose width is proportional to the
temperatures on the two sides. In this region, expectation values have a
different temperature dependence and show smooth peaks as a function of
$\zeta$. We explicitly compute the universal function describing such peaks. In
the specific case of interacting integrable models, our predictions are
analytically recovered by the generalized hydrodynamic approach.
