The Earth cannot drop to Absolute Zero because the universe would keep it above 2.7 K. Without the Sun, the only energy to the surface of the Earth is the heat from it's own interior. All you have to do is go down a kilometer or two and you are already up to a temperature of 140 F or more where deep diamond miners work. If the Sun went away, this would change only a slight amount as the Earth re-adjusts to a heat flow where the outer surface is no longer warmed by the Sun. My guess is that this heat flow is not enough to keep the earth above the freezing point of water, and that after perhaps a month or so, the latent solar heat stored in the oceans and crust would be exhausted. The temperature would stabilize probably somewhere below 200 K and be maintained thereafter by the heat flux from the Earth's interior for a few billion years.