  In ordinary materials, electrons conduct both electricity and heat, where
their charge-entropy relations observe the Mott formula and the Wiedemann-Franz
law. In topological quantum materials, the transverse motion of relativistic
electrons can be strongly affected by the quantum field arising around the
topological fermions, where a simple model description of their charge-entropy
relations remains elusive. Here we report the topological charge-entropy
scaling in the kagome Chern magnet TbMn$_6$Sn$_6$, featuring pristine Mn kagome
lattices with strong out-of-plane magnetization. Through both electric and
thermoelectric transports, we observe quantum oscillations with a nontrivial
Berry phase, a large Fermi velocity and two-dimensionality, supporting the
existence of Dirac fermions in the magnetic kagome lattice. This quantum magnet
further exhibits large anomalous Hall, anomalous Nernst, and anomalous thermal
Hall effects, all of which persist to above room temperature. Remarkably, we
show that the charge-entropy scaling relations of these anomalous transverse
transports can be ubiquitously described by the Berry curvature field effects
in a Chern-gapped Dirac model. Our work points to a model kagome Chern magnet
for the proof-of-principle elaboration of the topological charge-entropy
scaling.
