We explore a hadron resonance gas model using Tsallis non-extensive distribution to study the energy dependence of the product of the moments, Sσ and κσ² of net-proton multiplicity distributions of published STAR data in Au+Au collisions at relativistic heavy-ion collider (RHIC) energies. While excellent agreements are found between model predictions and measurements of Sσ and κσ² of most peripheral collisions and Sσ of most central collisions, the κσ² for most central collisions deviates significantly from the predictions particularly at √s_NN= 19.6 GeV and 27 GeV. This could be an indication of the presence of dynamical fluctuations, which are not contained in the HRG-Tsallis model.