Stefan-Boltzmann's law indicates that far-field blackbody radiation scales at the fourth power of temperature. The temperature dependence of radiative heat transfer in the near field is expected to be very different due to the contribution of evanescent waves. In this work, we experimentally observe such deviation on the radiative thermal conductance by bringing a hot micrometric sphere in the nearfield of a room-temperature planar substrate, down to a separation distance of few tens of nanometers. The influence of materials is assessed by using either SiO2 or graphite spheres, and SiO2, graphite and InSb substrates. Temperature differences as large as 900 K are imposed. A maximum near-field radiative thermal conductance of about 70 nW.K-1 is found for a graphite-graphite configuration. The experimental results demonstrate that the temperature exponent weakens in the near field, ranging from 2.2 to 4.1, depending on the gap distance and the materials. These results have broad consequences, in particular on the design of high-temperature nanoscale radiative energy devices.