Neutron stars are among the most enigmatic objects in the Universe. They possess the mass of our sun but are several billion times smaller than it. The matter in the cores of neutron stars is therefore compressed to densities that are several times higher than the density of atomic nuclei. Under such extreme physical conditions the conventional building blocks of matter as we know them (atoms, protons, electrons) give way to new and widely unexplored states of matter, such as superconducting quark matter and novel particle condensates searched for in the most powerful terrestrial collider experiments. In this paper we study the thermal evolution of neutron stars in order to explore the properties of ultradense matter and the inner workings of neutron stars. The calculations are performed in the framework of Einstein's theory of general relativity, since neutron stars curve the geometry of space-time so strongly that classical Newtonian theory of gravity fails to describe their properties.