  We revisit the effective parameter description of hot Brownian motion -- a
scenario where a colloidal particle is kept at an elevated temperature than the
ambient fluid. Due to the time scale separation between heat diffusion and
particle motion, a stationary halo of hot fluid is carried along with the
particle, resulting in a spatially varying comoving temperature and viscosity
profile. The resultant Brownian motion in the overdamped limit can be well
described by a Langevin equation with effective parameters such as effective
temperature $T_{\rm HBM}$ and friction coefficient $\zeta_{\rm HBM}$ that
quantifies the thermal fluctuations and the diffusivity of the particle. These
parameters can exactly be calculated using the framework of fluctuating
hydrodynamics. Additionally, it was also observed that configurational and the
kinetic degrees of freedom admits to different effective temperatures,
$T^{\mathbf{x}}_{\rm HBM}$ and $T^{\mathbf{v}}_{\rm HBM}$, respectively, with
the former predicted accurately from fluctuating hydrodynamics. A more rigorous
calculation by Falasco et. al. Physical Review E , 90, $032131(2014)$ extends
the overdamped description to a generalized Langevin equation where the
effective temperature becomes frequency dependent and consequently, for any
temperature measurement from a Brownian trajectory requires the knowledge of
this frequency dependence. We use this framework to expand on this earlier work
and look at the first order correction to the effective temperature. The
effective temperature is calculated from the weighted average of the
temperature field with the dissipation function. Further, we provide a closed
form analytical result for effective temperature in the small as well high
frequency limit and using this we determine the kinetic temperature from the
generalized Langevin equation and the Wiener-Khinchine theorem.
