Sufficient conditions for asymptotic lognormality of exact and approximate, unconditional and conditional $P$-values are established. It is pointed out that the mean, which is half the Bahadur slope, and the standard deviation of the asymptotic distribution of the log transformed $P$-value together, but not the mean alone, permit approximation of both the level and power of the test. This provides a method of discriminating between tests that have Bahadur efficiency one. The asymptotic distributions of the log transformed $P$-values of the common one- and two-sample tests for location are derived and compared.
Diane Lambert. W. J. Hall. "Asymptotic Lognormality of $P$-Values." Ann. Statist. 10 (1) 44 - 64, March, 1982. https://doi.org/10.1214/aos/1176345689