GaN Schottky diode rectifiers with reverse breakdown (V-RB) > 2 kV were fabricated on epitaxial layers grown on sapphire substrates. The temperature dependence of V-RB and forward turn-on voltage (V-F) were measured. The V-RB values display a negative temperature coefficient (-0.92 V .K-1 for 25-50 degreesC; -0.17 V .K-1 for 50-150 degreesC), indicative of surface- or defect-assisted breakdown. The V-F values decrease with increasing temperature. The room temperature breakdown voltage is approximately a factor of three lower than the theoretical maximum expected based on avalanche breakdown, and the current performance of GaN rectifiers is comparable to that of Si at the same on-resistance.