  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 $T_c$ for Bose-Einstein condensation. These results show
that the properties of the impurity vary strongly with temperature. In
particular, the energy exhibits a non-monotonic behavior close to $T_c$, and
the damping rises sharply close to $T_c$. We argue that this behaviour 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.
