Two-phase Cryogenic Avalanche Detectors (CRADs) with THGEM multipliers, optically read out with Geiger-mode APDs (GAPDs), were proposed as potential technique for charge recording in rare-event experiments. In this work we report on the degradation of the GAPD performance at cryogenic temperatures revealed in the course of the study of two-phase CRAD in Ar, with combined THGEM/GAPD-matrix multiplier; the GAPDs recorded secondary scintillation photons from the THGEM holes in the Near Infrared. The degradation effect, namely the loss of the GAPD pulse amplitude, depended on the incident X-ray photon flux. The critical counting rate of photoelectrons produced at the 4.4 mm2 GAPD, degrading its performance at 87 K, was estimated as 10000 per second. This effect was shown to result from the considerable increase of the pixel quenching resistor of this CPTA-made GAPD type. Though not affecting low-rate rare-event experiments, the observed effect may impose some limitations on the performance of CRADs with GAPD-based optical readout at higher-rate applications.