  We derive generalized Langevin equations for the translational and rotational
motion of a heated Brownian particle from the fluctuating hydrodynamics of its
non-isothermal solvent. The temperature gradient around the particle couples to
the hydrodynamic modes excited by the particle itself so that the resulting
noise spectrum is governed by a frequency-dependent temperature. We show how
the effective temperatures at which the article coordinates and (angular)
velocities appear to be thermalized emerge from this central quantity.
