We present some general considerations on the properties of a two-component ultracold Fermi gas along the BEC-BCS crossover. It is shown that the interaction energy and the free energy can be written in terms of a single dimensionless function h͑ , ͒, where =−͑k F a s ͒ −1 and = T / T F . The function h͑ , ͒ incorporates all the many-body physics and naturally occurs in other physical quantities as well. In particular, we show that the average rf-spectroscopy shift ␦͑ , ͒ and the molecular fraction f c ͑ , ͒ in the closed channel can be expressed in terms of h͑ , ͒ and thus have identical temperature dependence. The conclusions should have testable consequences in future experiments.