  The predictions from a non-extensive self-consistent theory recently proposed
are investigated. Transverse momentum ($p_T$) distribution for several hadrons
obtained in $p+p$ collisions are analyzed to verify if there are evidence for a
limiting effective temperature and a limiting entropic factor. In addition, the
hadron-mass spectrum proposed in that theory is confronted with available data.
  It turns out that all $p_T$-distributions and the mass spectrum obtained in
the theory are in good agreement with experiment with constant effective
temperature and constant entropic factor. The results confirm that the
non-extensive statistics plays an important role in the description of the
termodynamics of hadronic systems, and also that the self-consistent principle
holds for energies as high as those achieved in the LHC. A discussion on the
best $p_T$-distirbution formula for fitting experimental data is presented.
