Optically trapped two-component Fermi gases near a broad Feshbach resonance exhibit universal thermodynamics, where the properties of the gas are independent of the details of the two-body scattering interactions. We show that the measured cloud profiles of both species determine the total energy of universal gases for any trapping potential U and any spin mixture, without assuming either the local density approximation or harmonic confinement. A global proof of the virial theorem yields the total energy of the gas at all temperatures in the scale-invariant form E= ?U/ , where ? is an arbitrary energy scale in terms of which all length and energy scales that appear in the confining potential are written. This result enables model-independent energy measurement in any trap by observing only the cloud profiles, and provides a consistency check for many-body calculations in the universal regime. © 2008 The American Physical Society.