A set of generalized-BCS equations (GBCSEs) was recently derived from a temperature-dependent Bethe-Salpeter equation and shown to deal satisfactorily with the experimental data comprising the T_cs and the multiple gaps of a variety of high-temperature superconductors (SCs). These equations are formulated in terms of the binding energies W₁(T),W₂(T),… of Cooper pairs (CPs) bound via one- and more than one-phonon exchange mechanisms; they contain no direct reference to the gap/s of an SC. Applications of these equations so far were based on the observation that for elemental SCs |W₀₁|=△₀ at T = 0 inthe limit of the dimensionless BCS interaction parameter λ→0. Here △₀ is the zero-temperature gap whence it follows that the binding energy of a CP bound via one-phonon exchanges at T = 0 is 2|W₀₁|. In this note we carry out a detailed comparison between the GBCSE-based W₁(T) and the BCS-based energy gap △(T) for all 0≤T≤T_c and realistic, non-vanishingly-small values of λ. Our study is based on the experimental values of T_c Debye temperature , and ?₀ of several selected elements including the “bad actors” such as Pb and Hg. It is thus established that the equation for W₁(T) provides a viable alternative to the BCS equation for △(T). This suggests the use of, when required, the equation for W₂(T) which refers to CPs bound via two-phonon exchanges, for the larger of the two T-dependent gaps of a non-elemental SC. These considerations naturally lead one to the concept of T-dependent interaction parameters in the theory of superconductivity. It is pointed out that such a concept is needed both in the well-known approach of Suhl et al. to multi-gap superconductivity and the approach provided by the GBCSEs. Attention is drawn to diverse fields where T-dependent Hamiltonians have been fruitfully employed in the past.