  Dissipative quantum Rabi System, a finite-component system composed of a
single two-level atom interacting with an optical cavity field mode, exhibits a
quantum phase transition, which can be exploited to greatly enhance the
estimation precision of unitary parameters (frequency and coupling strength).
Here, using the quantum Langevin equation, standard mean field theory and
adiabatic elimination, we investigate the quantum thermometry of a thermal bath
surrounding the atom with quantum optical probes. With the increase of coupling
strength between the atom and the cavity field, two kinds of singularities can
be observed. One type of singularity is the exceptional point (EP) in the
anti-parity-time (anti-$\mathcal{PT}$) symmetrical cavity field. The other type
of singularity is the critical point (CP) of phase transition from the normal
to superradiant phase. We show that the optimal measurement precision occurs at
the CP, instead of the EP. And the direct photon detection represents an
excellent proxy for the optimal measurement near the CP. In the case where the
thermal bath to be tested is independent of the extra thermal bath interacting
with the cavity field, the estimation precision of the temperature always
increases with the coupling strength. Oppositely, if the thermal bath to be
tested is in equilibrium with the extra bath interacting with the cavity field,
noises that suppress the information of the temperature will be introduced when
increasing the coupling strength unless it is close to the CP.
