## 1 Introduction
The pioneers of this direction are the Japanese groups who put forward the project of the first gravitational-wave interferometer with cryogenic mirrors [14] at the end of the 20 th century. This project was then tested in the pilot version [15], and is currently implemented as an underground cryogenic interferometer KAGRA [16; 17]. To combat thermal noise, quantum photon noise and light pressure noise in a KAGRA setup, large heavy sapphire mirrors on sapphire suspensions [18] with low mechanical losses [19] in the coating films are used. At the same time, mirrors weighing tens of kilograms working under the influence of powerful optical pumping are suspended as pendulums for effective isolation from the background of seismic noise. This is required to preserve the high mechanical quality feature of mirrors and suspensions. Under such conditions, the problem of cooling mirrors to low temperatures becomes a serious problem due to poor thermal contact. In particular, when cooling suspended quartz mirrors through the exchange gas in the presence of a luminous flux of about 20 mW, the temperature of about 12-14 K has been reached in a month [20]. In contrast to the reference FP cavities required in the technique of optical frequency standards, the finesse of gravitational-wave interferometers with a kilometer base are very low (no more than 500), but their mirrors must withstand the high-power optical radiation (tens of kilowatts or more). Such requirements are also laid down in the European project Einstein Telescope (ET) [11] and in the development projects of third-generation interferometers LIGO [12]. LIGO plans to switch to silicon mirrors with cooling to 120 K [21]. The ET project is still considering this possibility along with other options [22]. The search continues for various materials for test mass mirrors and their spectrally selective reflective coatings. Their evolution under conditions of long-term operation in gravitational-wave bursts monitoring is investigated. In a recent paper by the KAGRA [23] group, the effect of changing the reflectivity of sapphire mirrors due to the adsorption of residual gas molecules (H\({}_{2}\)O molecules) on the cooled surface of mirrors is studied. An auxiliary measuring FP resonator with a finesse of about 50 000 is used. This increased finesse is required to register small changes in the reflectivity of the mirrors as a result of adsorption. The evaluation of the finesse is carried out by measuring the optical relaxation time of the Fabry-Perot resonance when the optical power is turned off. In the observation interval of 35 days with a vacuum of \(6\times 10^{-6}\) Pa, a drop in the finesse value of up to 5% is recorded (the growth rate of the adsorbed layer is estimated to be \(27\pm 1.9\) nm/day). In future gravitational-wave interferometers of the third generation, this effect should apparently be taken into account (for example, periodically destroying the adsorbed layer by a pulsed increase in optical pumping [23]), but in a number of other applications this effect can be neglected. In particular, it might be valid for solid-state resonance gravitational antenna of moderate sensitivity with an optical read out system, named as OGRAN detector [24]. In this case, a partial improvement of the thermal contact between the detector and mirrors is possible [25], but it cannot be complete in view of the necessity to preserve a high quality factor of the acoustic modes of mirrors [26; 2]. In addition, the mechanical and optical stability of the mirror's multilayer reflecting coating under power laser pump also becomes a serious problem (despite a very low absorption coefficient) that has to be addressed experimentally. These objectives, under the general term of the \"interferometer with cryogenic mirrors\" are part of the research program for developing the third generation of gravitational wave antennas [11; 12]. In this article, we report the particular result of our experimental study on the behavior of FP optical resonators with mirrors on substrate of different materials.