  Recent theoretical work has shown that higher-order moments of the Berry
curvature, e.g., Berry curvature quadrupole or hexapole moments can produce the
leading order nonlinear anomalous Hall response (NLAH) in systems with special
magnetic point group symmetry. Recent experimental work has reported the
observation of the Berry curvature quadrupole-induced third-order NLAH (i.e.,
Hall voltage proportional to the third power of the external electric field)
from cryogenic conditions to room temperature in an epitaxially grown material
platform with broken time-reversal symmetry. In this paper, using semiclassical
Boltzmann formalism in the relaxation time approximation, we compute the Berry
curvature quadrupole-induced nonlinear anomalous thermal Hall and Nernst
coefficients in time-reversal broken systems. In systems where Berry curvature
monopole and dipole moments vanish by symmetry, our results predict the
behavior of the leading order transverse anomalous thermoelectric coefficients
proportional to the third power of the applied longitudinal temperature
gradient, and are guaranteed to exist in systems that have already exhibited
the third-order nonlinear anomalous Hall effect in recent experiments.
