Does using robust standard errors change effect size? I ran some linear regressions in R using lm, with an interaction term (cat x cat) as the predictor of interest (and also incorporating a covariate). To calculate the effect sizes of the interaction as a Cohen's d, I followed Jake Westfall's method as explained at http://jakewestfall.org/blog/index.php/2015/05/27/follow-up-what-about-uris-2n-rule/.
Specifically, in R, I ran:
RSS <- c(crossprod(model$residuals))
MSE <- RSS / length(model$residuals)
RMSE <- sqrt(MSE)
[code omitted to store the four cell means]
d=((A1-B1) - (A2-B2))/(2*RMSE)

But then I determined my data had some heteroscedasticity issues and redid my models with robust standard errors.
Specifically, I used the sandwich package to adjust my t- and p-values for reporting with:
coeftest(model, vcov. = vcovHC(model, type='HC3'))

And I reported the robust SEs with emmeans:
modelEM <- emmeans(model, ~MayoFile*TreatmentMDX,
                  vcov = sandwich::vcovHC(model,type='HC3'))

All fine and dandy, but now I'm stuck with a question I ought to know the answer to: Does the effect size change too now I'm using robust SEs? My gut says "No, the effect size is the same, what's changed is whether or not it's significant" - but I'm not entirely confident in that conclusion.
Thanks for reading!

The definition of effect size that you're using is relative to the standard error, so if you estimate the standard error in a different way (which you do, if you switch to robust), your estimated effect size will change accordingly.
As @whuber correctly wrote in a comment, this does not mean that the true underlying effect size has changed, and if your robust standard error is indeed a better estimator of the true standard error here (which may well be), then chances are you now have a better effect size estimator.
However, when reporting, obviously you'll have to report the estimate, as you can't know what the truth is.
