The microphysics of molecular hydrogen formation has an influence on galactic-scale star-formation rates over cosmic time. H2 is the cooling agent needed to initiate the cloud collapse regulating the star-formation efficiency. H2 formation is inefficient in the gas phase under typical interstellar conditions, requiring dust grain surfaces to act as catalysts. Small carbonaceous grains with sizes from roughly 4 to 100–200 Å, including polycyclic aromatic hydrocarbons (PAHs), have been shown to increase the H2 formation rates due to their large surface-to-volume ratios. H2 formation rates on PAHs were previously thought to reduce above temperatures of 50 K and H atom recombination was believed to be highly efficient only below 20 K. Until now, both laboratory experiments and theoretical modelling have suggested that H2 cannot form on grains with temperatures above 100 K. Here we report evidence, through direct laboratory measurements, of the highly efficient formation of H2 at temperatures up to 250 K on carbonaceous surfaces mimicking interstellar dust. By pushing their formation towards warmer temperatures, the H2 molecules could start contributing substantially to the cooling of warmer gas (temperatures of roughly 50–250 K). This will have a marked impact on our understanding of H2 formation in nearby galaxies and its efficiency in high-redshift galaxies where the Cosmic Microwave Background already pushes dust temperatures to more than 20 K.