Think there are two identical Hot bodies one is on Earth and other in empty space. Both are at same temperature, same mass and identical in all perspectives, now if these two start losing heat which one will reach the local room temperature faster.
Things to consider: The temperature of the space (when in shadow) is -270 C, Average room temp on earth is 20 degree centigrade. In the space, no gasses of any kind are present (Its pure vacuum).
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We all know there are three types of heat transfers
I) Conduction (Within or between solids)
II) Convention (In fluids like gases and liquids)
III) Radiation (No medium is required)
Among these, Conduction transfers heat fast, then followed by Convention and then by Radiation. Actually the radiation is the most efficient and very slow way of losing heat.
Now coming to the problem the mouse on the Earth can do all the tree types of heat emission Conduction, Convention, and Radiation. But the Mouse on the space has nothing around it to perform CONDUCTION or CONVENTION. The only thing it can do is RADIATE, which is more efficient and slow. So the heat lost in the space is very slow process compared to any thing on earth.