  Unlike the conventional (linear) anomalous Nernst effect, the non-linear
anomalous Nernst effect (NLANE) can survive in an inversion symmetry broken
system even in the presence of time-reversal symmetry. Using semiclassical
Boltzmann transport theory, we derive the general expression of the non-linear
anomalous Nernst coefficient as the second-order response function to the
applied temperature gradient. We find that the non-linear Nernst current, which
flows perpendicular to the temperature gradient even in the absence of a
magnetic field, arises due to the Berry curvature of the states near the Fermi
surface, and thus is associated with purely a Fermi surface contribution. We
apply these results to bilayer WTe$_2$, which is an inversion broken but time
reversal symmetric type-II Weyl semimetal supporting chiral Weyl fermions. By
tuning the spin-orbit coupling, we show that the sign of the NLANE can change
in this system. Together with the angular dependence, we calculate the
temperature and chemical potential dependencies of NLANE in bilayer WTe$_2$,
and predict specific experimental signatures that can be checked in
experiments.
