  We present results for the entire set of anomalous charge and heat transport
coefficients for metallic systems in the presence of a finite-temperature heat
bath. In realistic physical systems this necessitates the inclusion of
inelastic dissipation mechanisms; relatively little is known theoretically
about their effects on anomalous transport. Here we demonstrate how these
dissipative processes are strongly intertwined with Berry-curvature physics.
Our calculations are made possible by the introduction of a Caldeira-Leggett
reservoir which allows us to avoid the sometimes-problematic device of the
pseudogravitational potential. Using our formulas, we focus on the
finite-temperature behavior of the important anomalous Wiedemann-Franz ratio.
Despite previous expectations, this ratio is found to be non-universal as it
can exhibit either an upturn or a downturn as temperature increases away from
zero. We emphasize that this derives from a \textit{competition} between Berry
curvatures having different signs in different regions of the Brillouin zone.
We point to experimental support for these observations and for the behavior of
an alternative ratio involving a thermoelectric response which, by contrast,
appears to be more universal at low temperatures. Our work paves the way for
future theory and experiment, demonstrating how inelastic scattering at
non-zero temperature affects the behavior of all anomalous transport
coefficients.
