We investigate thermal transport in square ice, a two-dimensional analog of spin ice, exploring the role played by emergent magnetic monopoles in transporting energy. Using kinetic Monte Carlo simulations based on energy-preserving extensions of single-spin-flip dynamics, we explicitly compute the (longitudinal) thermal conductivity κ over a broad range of temperatures. We use two methods to determine κ : a measurement of the energy current between thermal baths at the boundaries, and the Green-Kubo formula, yielding quantitatively consistent values for the thermal conductivity. We interpret these results in terms of transport of energy by diffusion of magnetic monopoles. We relate the thermal diffusivity κ /C , where C is the heat capacity, to the diffusion constant of an isolated monopole, showing that the subdiffusive motion of the monopole implies κ /C vanishes at zero temperature. Finally, we discuss the implications of these results for thermal transport in three-dimensional spin ice, in spin-ice materials such as Dy2Ti2O7 and Ho2Ti2O7 , and outline some open questions for thermal transport in highly frustrated magnets.