  The Brownian motion of a hot nanoparticle is described by an effective Markov
theory based on fluctuating hydrodynamics. Its predictions are scrutinized over
a wide temperature range using large-scale molecular dynamics simulations of a
hot nanoparticle in a Lennard-Jones fluid. The particle positions and momenta
are found to be Boltzmann distributed according to distinct effective
temperatures $T_\mathrm{HBM}$ and $T_\mathrm{k}$ . For $T_\mathrm{HBM}$ we
derive a formally exact theoretical prediction and establish a generalised
Einstein relation that links it to directly measurable quantities.
