We report results of a study of the laser induced damage threshold (LIDT) behavior of ion beam sputtered HfO2/SiO2 multilayer coatings on Yb:YAG using 1-on-1 and N-on-1 test protocols. The tests were conducted at ambient, vacuum, and cryogenic conditions using 280 ps pulses at λ=1030nm. The 1-on-1 LIDT of antireflection (AR) stacks is found to be only slightly reduced under vacuum and cryogenic conditions, while that of high reflectivity (HR) stacks is insensitive to environmental conditions within the uncertainty of the measurements. Cryogenic N-on-1 tests show the LIDT of the HR coating is almost the same as in the 1-on-1 tests. Conversely, the cryogenic N-on-1 test of the AR coating shows damage at ∼13J/cm2, a fluence lower than the 20.4J/cm2 of 1-on-1 tests. The AR damage behavior is found to be affected by imperfections at the Yb:YAG surface. These findings show that high surface quality is required to increase energy extraction from active mirror laser amplifiers.