## 2 Methods  
The method presented in this work focuses on a sensitive determination of the CO\({}_{2}\) gas temperature in the room temperature region, that is why particularly temperature sensitive rovibrational CO\({}_{2}\) transitions are selected to be probed. CO\({}_{2}\) absorption spectra of the selected lines with frequencies within 2349 to 2352 cm\({}^{-1}\) as obtained from HITRAN simulations [15] are shown in Figure 1.  
The major absorption lines at about 2351.45 and 2349.92 cm\({}^{-1}\) are due to the R(2) and R(0) rotational transitions of the fundamental asymmetric stretching vibration (\((00^{0}1)\leftrightarrow(00^{0}0)\)), respectively. The lower-energy level of these transitions is the vibrational ground state of the CO\({}_{2}\) molecules. Absorption lines with minor intensity originate from the R(16)\(-\)R(21) rotational transitions of so-called \"hot bands\" of the asymmetric CO\({}_{2}\) stretching mode, where the initial states of the transitions are already excited in the bending mode (\((01^{1}1)\leftarrow(01^{1}0)\)). The lower-energy states of these hot band transitions are higher in energy than the fundamental ground state transitions by 773 to 846 cm\({}^{-1}\). The basic principle of our temperature measurements relies on the Boltzmann-like population of the fundamental and hot band transitions, while the hot bands increase in intensity with increasing temperature, the R(2) and R(0) transitions at 2351.45 and 2349.92 cm\({}^{-1}\) show  
Figure 1: Absorption spectra of CO\({}_{2}\) gas in arbitrary units (“arb. units”) at two different temperatures based on HITRAN simulations (5% CO\({}_{2}\), 95% N\({}_{2}\), \(p=101\),325 Pa). decreasing intensity with increasing temperature. This opposite temperature dependence is the key property for the high sensitivity of the method presented. To the best of our knowledge, the selected CO\({}_{2}\) transitions have not been used for QCL-based temperature sensing so far. For this purpose the transitions are particularly well suited, as (i) high temperature sensitivity can be achieved due to the opposite temperature dependence, (ii) cost-effective sensors with simple design may be developed as absorption line intensities are sufficient for direct absorption spectroscopy, and (iii) other atmospheric gases, such as H\({}_{2}\)O, CO, NO, NO\({}_{x}\), and hydrocarbons, have low interferences with the selected CO\({}_{2}\) absorption spectrum. Importantly, the frequencies of all selected transitions lie within the tuning range of commercial distributed feedback quantum cascade lasers (\(\sim\)2 cm\({}^{-1}\)). The fundamental scheme of the experimental setup is shown in Figure 2. A pulsed distributed feedback Peltier-cooled QCL (Alpes lasers, #sb6804) with a frequency tuning range of about 2349.5-2353.4 cm\({}^{-1}\) and a nominal average power of \(\sim\)1 mW is used to probe the CO\({}_{2}\) vibrational-rotational transitions mentioned above. The laser is operated by a commercial QCL driver system (Q-MACS embedded, neoplas control GmbH, Greifswald, Germany). The laser beam is collimated by a single-side anti-reflection-coated ZnSe lens and focused through a home-made heatable aluminum gas cell on a fast HgCdZnTe (MCT) detector (Q-MACS IRDM-1 GA neoplas control GmbH, Vigo PVI-4 TE-10.6\(\times\)1-TO8 detector element, bandwidth 3-200 MHz, wedged BaF\({}_{2}\) window). The gas cell comprises two 1 mm thick anti-reflection-coated sapphire windows and is equipped with a gas in- and outlet for loading with gas samples. The inner volume of the gas cell has a dimension of 8 \(\times\) 188 \(\times\) 48 mm\({}^{3}\), where \(d=8\) mm equals the laser beam path length in the gas cell. Heating is achieved by water conducted through pipes drilled in the housing of the gas cell and controlled by a thermostat (Julabo CF31). For reference temperature measurements, two Pt1000 sensors (class 1/3 DIN B) with an accuracy of \(\pm\)0.14 K and a response time, \(t_{90}\), of \(\leq\)4 s are mounted close to the center of the gas cell.