In the present paper, the effective mass and energy of kaon in dense and hot nuclear matter is studied. Our special attention is paid to the comparison of various definitions of quasi-particle model of kaons used in transport theories. The effective mass of kaon defined in popular mean field theory increases with increasing of the density of nuclear matter, while the quantity used in covariant kaon dynamics decreases with the density increasing. The properties of mean field acting on kaon by hot and dense nuclear matter depends not only on the form of quasiparticle model of kaons, but on the motion of nucleons around it.