  The Brownian motion of a particle hotter than its environment is an iconic
out-of-equilibrium system. Its study provides valuable insights into nanoscale
thermal effects. Notably, it supplies an excellent diagnosis of thermal effects
in optically levitated particles, a promising platform for force sensing and
quantum physics tests. Thus, understanding the relevant parameters in this
effect is critical. In this context, we test the role of particles' shape and
material, using optically levitated nanodiamonds hosting NV centers to measure
the particles' internal temperature and center-of-mass dynamics. We present a
model to assess the nanodiamond internal temperature from its dynamics,
adaptable to other particles.
  We also demonstrate that other mechanisms affect the nanodiamond dynamics and
its stability in the trap. Finally, our work, by showing levitating
nanodiamonds as an excellent tool for studying nano-thermal effects, opens
prospects for increasing the trapping stability of optically levitated
particles.
