The random coding capacity of the Gaussian arbitrarily varying channel (GAVC) under a maximal probability of error criterion is equal to that of an additive white Gaussian noise (AWGN) channel with the interference power as additional noise. The deterministic coding capacity under an average probability of error criterion is zero if the transmitter power does not exceed the interference power. We show that the random coding capacity for average probability error is the same as that for maximal probability of error and that the randomization need only be over a sub-exponential number of codebooks. The achievable error exponent is related to the amount of randomization. An application of these results to the degraded broadcast channel is discussed.