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Enhancing of sensitivity of the opto-acoustical gravitational wave (GW) antenna OGRAN installed at the underground facilities of Baksan Neutrino Observatory is analyzed. Calculations are presented, showing a sensitivity improvement by two orders of magnitude after cooling the solid acoustical part of the antenna to the nitrogen temperature. The possibility of keeping the same optical scheme of the antenna at low temperature is discussed. The design of an upgraded construction for a cryogenic version of the OGRAN antenna is described. Test experiments with a cooled pilot model carrying cryogenic mirrors illuminated by an optical pump up to 0.5Ware presented.