The Rosette molecular cloud is promoted as the archetype of a triggered star-formation site. This is mainly due to its morphology, because the central OB cluster NGC 2244 has blown a circular-shaped cavity into the cloud and the expanding HII-region now interacts with the cloud. Studying the spatial distribution of the different evolutionary states of all star-forming sites in Rosette and investigating possible gradients of the dust temperature will help to test the 'triggered star-formation' scenario in Rosette. We use continuum data obtained with the PACS (70 and 160 micron) and SPIRE instruments (250, 350, 500 micron) of the Herschel telescope during the Science Demonstration Phase of HOBYS. Three-color images of Rosette impressively show how the molecular gas is heated by the radiative impact of the NGC 2244 cluster. A clear negative temperature gradient and a positive density gradient (running from the HII-region/molecular cloud interface into the cloud) are detected. Studying the spatial distribution of the most massive dense cores (size scale 0.05 to 0.3 pc), we find an age-sequence (from more evolved to younger) with increasing distance to the cluster NGC 2244. No clear gradient is found for the clump (size-scale up to 1 pc) distribution. The existence of temperature and density gradients and the observed age-sequence imply that star formation in Rosette may indeed be influenced by the radiative impact of the central NGC 2244 cluster. A more complete overview of the prestellar and protostellar population in Rosette is required to obtain a firmer result.Read less <