In this paper we prove a rate of convergence result for the density of normalized sample maxima to the appropriate limit density. Also a local limit theorem--with rates of convergence--for maxima of i.i.d. random variables is proved.
E. Omey. "Rates of Convergence for Densities in Extreme Value Theory." Ann. Probab. 16 (2) 479 - 486, April, 1988. https://doi.org/10.1214/aop/1176991768