Great question! The moon does still give off heat, though it gives off much less than a geologically “active” planet like Earth. Astronauts in the Apollo missions (Apollo 15 and 17) have actually measured heat flow at two different locations on the moon. The heat flow describes how fast an area on the surface of the moon is losing its heat.

The average heat flow of the surface of the moon is 18mWm^{- 2}, compared to 87 mWm^-2 on Earth (Francis & Oppenheimer, 2004). This means that Earth is losing heat almost five times faster than the moon. Most of the heat is probably lost as infrared radiation, a form of light that the human eye can’t see but that we can measure with special instruments.

Most of the Earth’s internal heat comes from the decay of radioactive isotopes of the elements potassium, uranium, and thorium. Radioactive decay produces a lot of energy (heat); that’s why we use radioactive material in electricity producing reactors. The budget of these elements is still high in Earth, so the planet is relatively hot. This is why we have plate tectonics and volcanoes. The moon has a similar concentration of these elements to the Earth’s mantle, but it is smaller, so the total amount of these elements is less. The moon was also hot early in its life, but it cooled much faster than the Earth because it is smaller. As spheres (the moon and earth are roughly sphere- shaped) get smaller, their surface area/volume ratio gets bigger and bigger and this makes them cool faster. You should calculated the surface area and volume of a few spheres of different size and see how the ratio s.a./v changes as they get larger and smaller.

Francis, P., Oppenheimer, C. (2004). Volcanoes: 2nd edition. New York: Oxford University Press.