  In this work, we study the thermodynamics of a two-level system (qubit)
embedded in a finite-temperature structured-bath under periodical measurements.
The system under measurements will reach a quasi-steady state, whose effective
temperature can be maintained lower than that of the surrounding environment.
To study the influence of the environmental oscillators from different regimes
of frequency on the qubit, the spectrum of the bath consisting of a large
number of bosonic harmonic oscillators can be approximately divided into three
parts according to their effects of cooling or heating. Due to the spectral
analysis over the structured-bath based on the non-Markovian master equation
beyond the rotating-wave approximation, we propose a sufficient cooling
condition for the bath in the context of quantum non-selective measurement. It
is consisted of two items: (i) the logarithmic derivative of the spectrum
around the system transition frequency is large enough, at least larger than
one half of the inverse temperature of the bath; (ii) the spectrum should have
a sharp high-frequency cutoff that is not far-detuning from the system
transition frequency. From this condition, we find that two popular types of
spectra, i.e., the modified Lorentzian models and the super-Ohmic models, are
available environments for cooling the open quantum system.
