The Rosette molecular cloud complex is a well-known Galactic star forming region with a morphology pointing towards triggered star formation. The distribution of its young stellar population and the gas properties question whether star formation could be globally triggered in the region. We focus on the characterisation of the most massive pre- and protostellar cores distributed throughout the molecular cloud to help the understanding of the star formation processes in the region. We observed a sample of 33 dense cores, identified in Herschel continuum maps, with the Effelsberg 100-m telescope. Using NH_3_ (1,1) and (2,2) measurements we characterise the dense core population computing rotational and gas kinetic temperatures and NH3 column density with multiple methods. We also estimate the gas pressure ratio and virial parameters to examine the stability of the cores. Using results from Herschel data we examine possible correlations between gas and dust parameters. Ammonia emission is detected toward 31 out of the 33 selected targets. We estimate kinetic temperatures between 12 and 20K, and column densities within the 10^14^-2x10^15^cm^-2^ range in the selected targets. The virial analysis suggests that most sources are likely to be gravitationally bound, while the linewidths are dominated by non-thermal motions. Our results are compatible with large scale dust temperature maps suggesting that the temperature decreases and column density increases with distance from NGC 2244 except for the densest protoclusters. We also identify a small spatial shift between the ammonia and dust peaks in the regions most exposed to irradiation from the nearby NGC 2244 stellar cluster. However, we find no trends in terms of core evolution with spatial location, in the number of prestellar to protostellar core ratio or the virial parameter. Star formation is more likely based on the primordial structure of the cloud in spite of the impact of irradiation from the nearby cluster, NGC 2244. The physical parameters from the NH_3_ measurements suggest gas properties in between those of low- and high-mass star forming regions, suggesting that the Rosette molecular cloud could host on-going intermediate mass star formation, and is unlikely to form high-mass stars.