We study the properties of K and K mesons in nuclear matter at finite temperature from a chiral unitary approach in coupled channels that 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. We calculate K and K (off-shell) spectral functions and single-particle properties. The K effective mass gets lowered by about -50 MeV in cold nuclear matter at saturation density and by half this reduction at T=100 MeV. The p-wave contribution to the K optical potential, due to {lambda},{sigma}, and {sigma}* excitations, becomes significant for momenta larger than 200 MeV/c and reduces the attraction felt by the K in the nuclear medium. The K spectral function spreads over a wide range of energies, reflecting the melting of the {lambda}(1405) resonance and the contribution of YN{sup -1} components at finite temperature. In the KN sector, we find that the low-density theorem is a good approximation for the K self-energy close to saturation density due to the absence of resonance-hole excitations. The K potential shows a moderate repulsive behavior, whereas the quasiparticle peak is considerably broadened with increasing density andmore » temperature. We discuss the implications for the decay of the {phi} meson at GSI Schwerionen Synchrotron energies as well as in the future Facility for Antiproton and Ion Research project.« less