Greenwood's approximation for the thermal resistance of a cluster of microcontacts is used recursively to estimate the thermal resistance due to a fractal array of circular contact areas motivated by Archard's contact model. The results are then extended to the case of sliding contacts, using a technique due to Burton. It is found that the total resistance converges on a limit when arbitrarily large numbers of fractal scales are included, but the fine scale features in the contact area have a disproportionate effect at high Peclet number and hence reduce the proportion of frictional heating passing into the moving body. (C) 2010 Elsevier Ltd. All rights reserved.