The delivery of high RF power---from hundreds of kW to MW---by klystrons, is linked with a high overall energy consumption. A research programme led by CERN in collaboration with the industry is being conducted to understand what limits klystron efficiency and how to develop high-efficiency klystrons. As a result of this program, two first prototypes of X-band (11.994 GHz) high-efficiency klystrons have been successfully designed and manufactured in collaboration with Canon Electron Tubes and Devices. The first results look promising, revealing a remarkable ~60% efficiency, and validating the proposed HE klystron technology. A comprehensive characterisation campaign has been conducted at CERN to verify and demonstrate these results. The methodology for the HEK tubes characterisation is based in two independent measurements: a RF power measurement, and a calorimetric methodology ---less subject to calibration inaccuracies. We describe the setups, principle of the calorimetry methodology, and we discuss the feasibility and precision of the results.