In this work, we study for the first time the thermal evolution of twin star pairs, i.e., stars that present the same mass but different radius and compactness. We collect available equations of state that give origin to a second branch of stable compact stars with quarks in their core. For each equation of state, we investigate the particle composition inside stars and how differently each twin evolves over time, which depends on the central density/pressure and consequent crossing of the threshold for the Urca cooling process. We find that, although the general stellar thermal evolution depends on mass and particle composition, withing one equation of state, only twin pairs that differ considerably on compactness can be clearly distinguished by how they cool down.