Spectroscopic observations of distant quasars have resulted in the detection of molecular hydrogen in intervening damped Lyman alpha absorption clouds (DLAs). We use observations compiled from different experimental groups to show that the molecular hydrogen abundance exhibits a dramatic increase over a cosmological time period corresponding to 13 to 24 per cent of the age of the Universe. We also tentatively show that the heavy-element abundances in the same gas clouds exhibit a faster and more well-defined cosmological evolution compared with the general DLA population over the same time baseline. We argue that this latter point is unsurprising, because the general DLA population arises in a wide variety of galaxy types and environments, and thus a spans broad range of ISM gas-phases and abundances at the same cosmic time. DLAs exhibiting H-2 absorption may therefore circumvent this problem, efficiently identifying a narrower class of objects, and provide a more sensitive probe of cosmological chemical evolution.