  We present a theory of radio-frequency spectroscopy of impurities interacting
with a quantum gas at finite temperature. By working in the canonical ensemble
of a single impurity, we show that the impurity spectral response is directly
connected to the finite-temperature equation of state (free energy) of the
impurity. We consider two different response protocols: "injection", where the
impurity is introduced into the medium from an initially non-interacting state;
and "ejection", where the impurity is ejected from an initially interacting
state with the medium. We show that there is a simple mapping between injection
and ejection spectra, which is connected to the detailed balance condition in
thermal equilibrium. To illustrate the power of our approach, we specialize to
the case of the Fermi polaron, corresponding to an impurity atom that is
immersed in a non-interacting Fermi gas. For a mobile impurity with a mass
equal to the fermion mass, we employ a finite-temperature variational approach
to obtain the impurity spectral response. We find a striking non-monotonic
dependence on temperature in the impurity free energy, the contact, and the
radio-frequency spectra. For the case of an infinitely heavy Fermi polaron, we
derive exact results for the finite-temperature free energy, thus generalizing
Fumi's theorem to arbitrary temperature. We also determine the exact dynamics
of the contact after a quench of the impurity-fermion interactions. Finally, we
show that the injection and ejection spectra obtained from the variational
approach compare well with the exact spectra, thus demonstrating the accuracy
of our approximate method.
