Will p-values vary more from study to study when the effect is small rather than large? Just want the title says. If you run ten studies with n = 50. Will the p-values vary more if examining an effect with Cohen's d of .1 vs .5?
 A: As whuber comments, in general the answer is yes. Here are the p values in a simulation of 100 experiments each with 25 participants in each one of two groups, a default t test and effect sizes 0.1, 0.5 and 0.8:

The standard deviations of the p values for the three effect sizes are 0.29, 0.24 and 0.07, respectively.
R code:
n_sim <- 1e2
n_sample_per_group <- 25
effect_sizes <- c(0.1,0.5,0.8)

p_values <- matrix(nrow=n_sim,ncol=length(effect_sizes))
for ( ii in 1:n_sim ) {
    set.seed(ii)    # for replicability
    for ( jj in seq_along(effect_sizes) ) {
        p_values[ii,jj] <- 
            t.test(x=rnorm(n_sample_per_group),y=rnorm(n_sample_per_group,effect_sizes[jj]))$p.value
    }
}

apply(p_values,2,sd)

beeswarm_matrix <- function(MM, amount=0.3, add.boxplot=FALSE, add.beanplot=FALSE, names=NULL, pt.col=NULL, ...) {  # beeswarm plots of matrix columns
    plot(c(1-2*amount,ncol(MM)+2*amount),range(MM,na.rm=TRUE),xaxt="n",type="n",...)
    axis(1,at=1:ncol(MM),labels=if(is.null(names)){colnames(MM)}else{names},...)
    if ( add.boxplot ) boxplot(MM, add=TRUE, xaxt="n", outline=FALSE, border="grey", ...)
    if ( add.beanplot ) {
        require(beanplot)
        sapply(1:ncol(MM),function(xx)beanplot(MM[,xx],add=TRUE,what=c(0,1,1,0),xaxt="n",
            col=c(rep("lightgray",3),"lightgray"),border=NA, at=xx,...))
    }
    pt.col.mat <- matrix(if(is.null(pt.col)){"black"}else{pt.col},nrow=nrow(MM),ncol=ncol(MM),byrow=TRUE)
    points(jitter(matrix(1:ncol(MM),nrow=nrow(MM),ncol=ncol(MM),byrow=TRUE),amount=amount),MM,col=pt.col.mat,...)
}

opar <- par(mai=c(1,.8,.1,.1))
    beeswarm_matrix(p_values,pch=19,add.beanplot=TRUE,las=1,names=effect_sizes,xlab="Effect size (Cohen's d)",ylab="p values")
par(opar)

