The prospect for developing a passively Q-switched Yb:YAG/Cr:YAG monolithic microchip laser that operates at cryogenic temperature is theoretically analyzed. It is concluded that such a system has the potential to deliver laser pulses with improved energy and increased peak power in comparison with composite Yb:YAG/Cr:YAG or Nd:YAG/Cr:YAG devices that are operated at room temperature. Consequently, a cryogenically cooled Yb:YAG/Cr:YAG system is built and the emission performances are investigated. Laser pulses with 3.2 mJ energy, 6.1 MW peak power and high beam quality of M2 = 1.8 are achieved. By increasing the pump beam diameter, laser pulses with higher energy 32 mJ are obtained at 25 MW peak power with M2 = 5.4. To our knowledge, these are the best results obtained from passively Q-switched composite Yb:YAG/Cr:YAG monolithic microchip lasers.