  In this paper, one discusses the effects of the Tsallis entropy on the radial
pressure distribution in the proton. Using a damped confinement potential the
pressure distribution is obtained from the Tsallis entropy approach, being the
entropic-index $\omega$ connected with the proton temperature concerning some
transition temperature. Then, the approach allows the study of the proton
thermal evolution up to the Quark-Gluon Plasma regime. The von Laue stability
condition, arising from the pressure distribution results in positive and
negative energy regions. An analogy between the results for the radial pressure
distribution and the proton-proton and the antiproton-proton total cross
section is performed. The negative energy region is identified with the odderon
exchange while the positive represents the pomeron exchange dominance above
some transition energy $\sqrt{s_c}$. The hollowness effect is also discussed in
terms of the results obtained and the proposed analogy.
