  We study the stationary dynamics of an active interacting Brownian particle
system. We measure the violations of the fluctuation dissipation theorem, and
the corresponding effective temperature, in a locally resolved way. Quite
naturally, in the homogeneous phases the diffusive properties and effective
temperature are also homogeneous. Instead, in the inhomogeneous phases (close
to equilibrium and within the MIPS sector) the particles can be separated in
two groups with different diffusion properties and effective temperatures.
Notably, at fixed activity strength the effective temperatures in the two
phases remain distinct and approximately constant within the MIPS region, with
values corresponding to the ones of the whole system at the boundaries of this
sector of the phase diagram. We complement the study of the globally averaged
properties with the theoretical and numerical characterization of the
fluctuation distributions of the single particle diffusion, linear response,
and effective temperature in the homogeneous and inhomogeneous phases. We also
distinguish the behavior of the (time-delayed) effective temperature from the
(instantaneous) kinetic temperature, showing that the former is independent on
the friction coefficient.
