We present the low-temperature Geiger-mode characteristics of GaN (gallium nitride) p-i-n avalanche photodiodes (APDs). The devices have a breakdown voltage of −95V and a temperature dependence of $\sim $ 0.0159 ± 0.0034 V/K near 300K. The room-temperature (300 K) dark-count rate (DCR) is 23.8 MHz for a $75\times 75\,\,\mu \text{m}^{2}$ device biased at 1 % overvoltage. The DCR halves when lowering the temperature by 50°C. Based on the temperature-dependent characteristics of the DCR, we identify band-to-band tunneling as the dominant DCR generation mechanism. At 4.65 % overvoltage and 375 nm, the photon detection efficiency (PDE) is 0.82 %- limited by a low breakdown probability of 1.7 %. We discuss the measurement setup and the method to extract count rates, which is based on the Poisson distribution of the time intervals between Geiger-mode breakdowns of the APD. The setup includes a custom circuit to bias the diode and amplify its signals, a steady ultraviolet (UV) light source, and a system to control the temperature of the APD with a thermoelectric element in the range from −40 to 20°C.