https://stats.stackexchange.com/questions/452/always-report-robust-white-standard-errors

It has been suggested by Angrist and Pischke that Robust (i.e. robust to heteroskedasticity or unequal variances) Standard Errors are reported as a matter of course rather than testing for it. Two questions:
What is impact on the standard errors of doing so when there is homoskedasticity?
Does anybody actually do this in their work?

Using robust standard errors has become common practice in economics. Robust standard errors are typically larger than non-robust (standard?) standard errors, so the practice can be viewed as an effort to be conservative.
In large samples (e.g., if you are working with Census data with millions of observations or data sets with "just" thousands of observations), heteroskedasticity tests will almost surely turn up positive, so this approach is appropriate.

Another means for combating heteroskedasticity is weighted least squares, but this approach has become looked down upon because it changes the estimates for parameters, unlike the use of robust standard errors. If your weights are incorrect, your estimates are biased. If your weights are right, however, you get smaller ("more efficient") standard errors than OLS with robust standard errors.

down vote
In Introductory Econometrics (Woolridge, 2009 edition page 268) this question is addressed. Woolridge says that when using robust standard errors, the t-statistics obtained only have distributions which are similar to the exact t-distributions if the sample size is large. If the sample size is small, the t-stats obtained using robust regression might have distributions that are not close to the t distribution and this could throw off inference.

Robust standard errors provide unbiased standard errors estimates under heteroscedasticity. There exists several statistical text books that provide a large and lengthy discussion on robust standard errors. The following site provides a somewhat comprehensive summary on robust standard errors:
https://economictheoryblog.com/2016/08/07/robust-standard-errors/
Coming back to your questions. Using robust standard errors is not without caveats. According to Woolridge (2009 edition, page 268) using robust standard errors, the t-statistics obtained only have distributions which are similar to the exact t-distributions if the sample size is large. If the sample size is small, the t-stats obtained using robust regression might have distributions that are not close to the t distribution. This could throw off inference. Furthermore, in case of homoscedasticity, robust standard errors are still unbiased. However, they are not efficient. That is, conventional standard errors are more precise than robust standard errors. Finally, using robust standard errors is common practice in many academic fields.