  The properties of $K$ and $\bar K$ mesons in nuclear matter at finite
temperature are obtained from a chiral unitary approach in coupled channels
which incorporates the $s$- and p-waves of the kaon-nucleon interaction. The
in-medium solution accounts for Pauli blocking effects, mean-field binding on
all the baryons involved, and $\pi$ and kaon self-energies. The $\bar K$
spectral function spreads over a wide range of energies, reflecting the melting
of the $\Lambda (1405)$ resonance and the contribution of hyperon-hole
components at finite temperature. In the $KN$ sector, the quasi-particle peak
is considerably broadened with increasing density and temperature. We also
study the energy weighted sum rules of the kaon propagator by matching the
Dyson form of the propagator with its spectral Lehmann representation at low
and high energies. The sum rules for the lower energy weights are fulfilled
satisfactorily and reflect the contributions from the different quasi-particle
and collective modes of the spectral function. We analyze the sensitivity of
the sum rules to the distribution of spectral strength and their usefulness as
quality tests of model calculations.
