My son had this theory that the icy comets cooled the hot earth during the late heavy bombardment. What do we know about this cooling effect on the early evolution of the earth? He intended his theory to apply to all "fiery" planets.
Nice hypothesis. However impacts from comets, thought they are made of ice won't cool it down, rather if the nucleus survives from the intense heat, and crashes down, it will convert a lot of kinetic energy into heat, more than what it could cool down.
However, the hypothesis is partly correct! The comets and asteroids have Asteroidal water, with either direct water, more often in comets, or precursors of $H_2O$. Now this lead to the formation of reservoirs in the craters, on our protoearth leading to the formation of water bodies, these water bodies effectively cooled down Earth's crust by the Latent heat of vaporization, which is quite similar to when the pan is hot, water cools it down. This evaporated into the air, which contributed on the atmosphere and once the temperature dropped due to radiative cooling, the water rained down to cool it down further (Rain would occur after the cooling till at least $100$ °C, since clouds can rain down after condensation).
Secondly, the ejecta plume released would contain a lot of aerosols which would block the sunlight from reaching the earth, which would block the sunlight from the solar irradiance. It is applicable for the only the protoearth and ejecta winter, since in present day, any such impact would release $CO_2$ into the atmosphere, increasing the greenhouse effect, also the water would not cool the Earth's crust down because it is already in thermal equilibrium, so it would not work in the present day scenario.