Within the framework of the Akyüz-Winther potential, we introduce a new form of the universal function of nuclear proximity formalism that is directly dependent on the surface diffuseness parameter. We apply this analytical formula in the formalism of the proximity potential 2010 (Prox. 2010) and try to reproduce the empirical data of the fusion barrier heights over a wide mass range of colliding nuclei. Our detailed study reveals that the best-fitting values for the diffuseness parameter are located in a range between a = 0.744 and 0.892 fm, which is considerably larger than the value of a 0.65 fm deduced from elastic-scattering data. It is found that the ion-ion potential due to the modified form of the Prox. 2010 explains the fusion barrier heights within ±3% on average. With the help of coupled-channels calculations, we calculate and compare with the available experimental data the fusion excitation functions. The results obtained show that the use of the new form of universal function, along with the fitted values of the diffuseness parameter, improves the agreement with the data in comparison with the original form of the Prox. 2010 potential. The present investigation suggests that the energy dependence of the nucleus-nucleus potential through the diffuseness parameter is needed to explain the sub-barrier fusion excitation functions. This means that the energy dependence plays a key role in the problem of the abnormally large diffuseness of the interaction potential

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