Experimental data on inclusive cross sections of particles produced in high energy proton-(anti)proton collisions at ISR, RHIC, and Tevatron are analyzed in the framework of z-scaling. New features of the scaling function ψ(z) are established. These are flavor independence of ψ(z) including particles with heavy flavor content and saturation at low z. The flavor independence means that the shape of the scaling function ψ(z) is the same for different hadron species. The saturation corresponds to flattening of ψ(z) for z < 0.1. Relations of model parameters used in data z-presentation with some thermodynamical quantities (entropy, specific heat, temperature) are discussed. It is shown that behavior of the particle spectra at low z is controlled by a parameter c interpreted as a specific heat of the created medium associated with production of the inclusive particle. The saturation regime of ψ(z) observed at low z is assumed to be preferable in searching for phase transitions of hadron matter and for study of non-perturbative QCD in high energy proton-(anti)proton collisions at U70, RHIC, Tevatron, and LHC.