The hyperbolic Pöschl-Teller potential is one of the few known solvable potentials in supersymmetric quantum mechanics that contains two parameters, and this potential’s parameters exhibit a cross-additivity change under the condition of shape invariance. This article explores how the two parameters’ cross-additivity affects the corresponding energy levels and wave functions. This paper uses supersymmetric quantum mechanics and potential algebra methods to accurately solve the energy eigenvalues and eigenwave functions corresponding to the hyperbolic Pöschl-Teller potential with the cross-additivity of two parameters. The results obtained by the two methods are consistent and correspond to each other. And the cross-additivity changes in parameters result in partial missing energy levels and wave functions of the hyperbolic Pöschl-Teller potential.