Heat transfer governs the bit size and writing rate during sub-micrometer thermomechanical data storage with Atomic Force Microscope (AFM) cantilevers. The present work predicts the temperature distribution and rates of heat flow in the AFM tip and the substrate as functions of the peak cantilever temperature, the diameter of the tip-substrate contact, and the thickness of the deforming polymer coating on the silicon substrate. The calculations consider increased phonon scattering, radiation losses, and gas conduction losses at the silicon tip boundaries. Nearly ballistic phonon transport in the tip augments the dependence of the heat rate into the polymer on the tip-polymer contact diameter. For a cantilever heater temperature of 700 K and a polymer layer thickness of 80 nm, the temperature at the tip-polymer interface is predicted for contact diameters from 4 nm to 50 nm. This work models the deformation of the polymer layer during data writing and predicts data bit size as a function of tip temperature and writing time. These simulations will help optimize the design of the cantilever and the polymer data layer, with the goal of increasing the spatial density and rate of bit formation.