Temperature dependence of effective ionization coefficients in GaAs is formulated as a single polynomial function of temperature, which allows analytical expressions for breakdown voltage of GaAs $p^+n$ junctions as a function of temperature. At 300 K, extracted effective ionization coefficient of GaAs $p^+n$ junction especially agrees well with the published result of <111> oriented GaAs. The analytic results agree with the simulation as well as the experimental ones reported within 10% in error for the doping concentrations in the range of $10_{14}cm_{-3}~10_{17}cm_{-3}$ at 100 K, 300 K and 500 K.