  The thermal Hall effect, which arises when heat flows transverse to an
applied thermal gradient, has become an important observable in the study of
quantum materials. Recent experiments found a large thermal Hall conductivity
$\kappa_{xy}$ in many high-temperature cuprate superconductors, including deep
inside the Mott insulator, but the underlying mechanism remains unknown. Here,
we uncover a surprising linear temperature dependence for the inverse thermal
Hall resistivity, $1/\rho_H=-\kappa_{xx}^2/\kappa_{xy}$, in the Mott insulating
cuprates $\mathrm{La_2CuO_4}$ and $\mathrm{Sr_2CuO_2Cl_2}$. We also find this
linear scaling in the pseudogap state of Nd-LSCO in the out-of-plane direction,
highlighting the importance of phonons. On the electron-doped side, the linear
inverse thermal Hall signal emerges in NCCO and PCCO at various dopings,
including in the strange metal. Although such dependence arises in the simple
Drude model for itinerant electrons, its origin is unclear in strongly
correlated Mott insulating or pseudogap states. We perform a Boltzmann analysis
for phonons that incorporates skew-scattering, and we are able to identify
regimes where a linear $T$ inverse Hall resistivity appears. Finally, we
suggest future experiments that would further our fundamental understanding of
heat transport in the cuprates, and other quantum materials.
