In Bayesian Data Analysis 3 (page 151) in the sub-section P-values and u-values
there is a part which reads:
In the special case that the parameters $\theta$ are known (or estimated to a very high precision) or in which the test statistic $T(y)$ is ancillary (that is, if it depends only on observed data and if its distribution is independent of the parameters of the model) with a continuous distribution, the posterior predictive p-value $\text{Pr}(T(y^{rep})>T(y) \mid y)$ has a distribution that is uniform if the model is true. Under these conditions, p-values less than 0.1 occur 10% of the time, p-values less than 0.05 occur 5% of the time, and so forth.
More generally, when posterior uncertainty in $\theta$ propagates to the distribution of $T(y|\theta)$, the distribution of the p-value, if the model is true, is more concentrated near the middle of the range: the p-value is more likely to be near 0.5 than near 0 or 1. (To be more precise, the sampling distribution of the p-value has been shown to be "stochastically less variable" than uniform.)
I don't understand the second paragraph and would greatly appreciate any further explanation on this point, or an example.
I think the first paragraph I've quoted refers to the "classical" p-value distribution under the null-hypothesis(?).
Gelman also has a paper on this topic which mentions the same point as above.