What is the mechanism for the reaction of an organic hydroperoxide with sulfuric acid?
As Marko has commented it is like the Hock rearrangement, which is the migratory step in the cumene process.

It is somewhat similar to the Beckmann rearrangement except that there is an oxygen instead of a nitrogen.




The acid protonates the peroxide, generating a great leaving group on electron-deficient oxygen.
Migration towards electron-deficient oxygen occurs with loss of water to give an oxonium ion.
The migratory aptitude here is determined in the same way as for the Baeyer-Villiger reaction. There is a buildup of positive charge on the migrating carbon during the migratory step, and therefore the group that can best stabilise the charge preferentially migrates. The methoxy-substituted ring is electron-rich and the positive charge is more stabilised on it than on the unsubstituted phenyl group. The methyl group is even worse at stabilising positive charge.
Under the dilute acid conditions this oxonium ion is hydrolysed.