  We theoretically investigate how quasi-particle properties of an attractive
Fermi polaron are affected by nonzero temperature and finite impurity
concentration. By applying both non-self-consistent and self-consistent
many-body $T$-matrix theories, we calculate the polaron energy (including decay
rate), effective mass, and residue, as functions of temperature and impurity
concentration. The temperature and concentration dependences are weak on the
BCS side with a negative impurity-medium scattering length. Toward the strong
attraction regime across the unitary limit, we find sizable dependences. In
particular, with increasing temperature the effective mass quickly approaches
the bare mass and the residue is significantly enhanced. At the temperature
$T\sim0.1T_{F}$, where $T_{F}$ is the Fermi temperature of the background Fermi
sea, the residual polaron-polaron interaction seems to become attractive. This
leads to a notable down-shift in the polaron energy. We show that, by taking
into account the temperature and impurity concentration effects, the measured
polaron energy in the first Fermi polaron experiment {[}A. Schirotzek
\textit{et al.}, Phys. Rev. Lett. \textbf{102}, 230402 (2009){]} can be better
theoretically explained.
