We study the properties of open-charm mesons ($D$ and $\bar {D}$) in nuclear matter at finite temperature within a self-consistent coupled-channel approach. The meson-baryon interactions are adopted from a type of broken SU(4) s-wave Tomozawa-Weinberg terms supplemented by an attractive scalar-isoscalar attraction. The in-medium solution at finite temperature incorporates Pauli blocking effects, mean-field binding on all the baryons involved, and $\pi$ and open-charm meson self-energies in a self-consistent manner. In the $DN$ sector, the $\Lambda_c$ and $\Sigma_c$ resonances, generated dynamically at 2593 MeV and 2770 MeV in free space, remain close to their free-space position while acquiring a remarkable width due to the thermal smearing of Pauli blocking as well as from the nuclear matter density effects. As a result, the $D$ meson spectral density shows a single pronounced peak for energies close to the $D$ meson free-space mass that broadens with increasing matter density with an extended tail particularly towards lower energies. The $\bar D$ potential shows a moderate repulsive behavior coming from the dominant I=1 contribution of the $\bar D N$ interaction. The low-density theorem is, however, not a good approximation for the $\bar D$ self-energy in spite of the absence of resonance-hole contributions close to threshold in this case. We speculate the possibility of $D$-mesic nuclei as well as discuss some consequences for the $J/\Psi$ suppression in heavy-ion collisions, in particular for the future CBM experiment at FAIR.