Strongly interacting Fermi gases are of great current interest. Not only are fermions the most common particles in the universe, but they are also thought to have a universal thermodynamic behavior for strong interactions \cite{heiselberg,carlson,ho}. Recent experiments on ultra-cold Fermi gases provide an unprecedented opportunity to test universality in the laboratory \cite{hara,thomas,partridge,stewart,luo}. In principle this allows - for example - the interior properties of hot, dense neutron stars to be investigated on earth. Here we carry out a detailed test of this prediction. We analyze results from three ultra-cold fermion experiments involving two completely distinct atomic species in different kinds of atomic trap environments \cite{partridge,stewart,luo}. The data is compared with the predictions of a recent strong interaction theory \cite{hldepl,hldpra}. Excellent agreement is obtained, with no adjustable parameters. By extrapolating to zero temperature, we show that the experimental measurements yield a many-body parameter} $\mathbf{\beta\simeq-0.59\pm.07}$, \textbf{describing the universal energy of strongly interacting Fermi gases.