Grand Unified Theories (GUTs) provide a possible solution to the long-standing cosmological problem of the apparent asymmetry between matter and antimatter1-9. We have argued recently10 that in a class of GUTs one may relate the baryon asymmetry to certain contributions to the neutron electric dipole moment, dn. In the absence of cancellations due to an unforeseen symmetry, we thereby obtain a qualitative cosmological lower limit on dn. We emphasize here that the experimental upper limit on dn and the present cosmological baryon-to-photon ratio (nB/nγ) can be used to limit the amount of extra entropy generated after baryosynthesis. This restriction on entropy generation becomes quite severe if, as we argue here on the basis of explicit GUTs, d n is actually considerably larger than our previous cosmological lower bound. As an application of our restriction of entropy generation, we establish an upper limit on shear in the early Universe.