We consider the quasi-static mechanical response of microphase-segregated A-B random block copolymers to compressive strains in the strong segregation limit. Our analytical calculations show that the elastic constant that describes this response exhibits a nonmonotonic dependence on the fraction of blocks of type A (for a given degree of blockiness). Furthermore, the magnitude of the elastic constant is extremely sensitive to the disorder correlations along the copolymer backbones. The physical origin of the variation of the elastic constant with block sequence distribution and the size and stiffness of the blocks is discussed.