Thermodynamical stability of different variants of FePt nanoparticles has been studied using DFT molecular dynamics. The melting temperature and general stability at elevated temperatures have been estimated from both energy difference and atomic root-mean-square displacement functions. The investigated systems include multi-shell nanoparticles of iron and platinum with icosahedral symmetry and a magic number of atoms (55): iron-terminated Fe43Pt12 and platinum-terminated Fe12Pt43. Additionally, the cuboctahedral Fe24Pt31 particle, cleaved-out of the bulk structure, has been studied using the same procedure. Molecular dynamics simulations have been performed for a range of temperatures reaching above the melting points. The calculations confirmed high structural instability of the Fe-terminated nanoparticles and a strong stabilizing effect of the Pt-termination in the shell-type icosahedral particles. The advantage of the presented study is the self-consistency of the nanoparticle band structure in each time step, including magnetic interactions among local magnetic moments.