We consider a mobile impurity immersed in a Bose gas at finite temperature. Using perturbation theory valid for weak coupling between the impurity and the bosons, we derive analytical results for the energy and damping of the impurity for low and high temperatures, as well as for temperatures close to the critical temperature Tc for Bose-Einstein condensation. These results show that the properties of the impurity vary strongly with temperature. In particular, the energy exhibits a nonmonotonic behavior close to Tc, and the damping rises sharply close to Tc. We argue that this behavior is generic for impurities immersed in an environment undergoing a phase transition that breaks a continuous symmetry. Finally, we discuss how these effects can be detected experimentally.