The stiffness of thermal transport in ELMy H-modes is explored in a series of carefully chosen JT-60U plasmas and with temperature predictions based on several transport models. Four scans of pedestal temperature, Tped, with constant heating power and one heating power scan with constant Tped are presented. We find that 3080% increases in Tped are associated with 10-70% increases in core temperature even though the total heating power is constant. Increasing the heating power by 45% gives almost the same core temperatures (and a 12% density increase) in a group of five plasmas with the same pedestal temperature. The results can be characterized as having relatively soft transport in the plasma periphery and relatively stiff transport in the core. Another series of experiments varied the heating in the deep core by employing different groups of neutral beams that deposit their power on-axis and off-axis. In these plasmas on-axis heating produces systematically more peaked temperature profiles; the rise from the periphery to the central region is ~20% higher in plasmas that have 60% more heating power inside r=a/2. Transport models are tested by solving the power balance equations to predict temperatures, which are then compared to the measurements. The RLWB and IFS/PPPL models predictions generally agree with the measured temperatures, but the Multimode model uniformly predicts temperatures that are too high except for the central sawtoothing region.