We consider a generalized statistical mechanics model for the creation process of cosmic rays which takes into account local temperature fluctuations. This model yields Tsallis statistics for the cosmic ray spectrum. It predicts an entropic index q given by q=11/9 at largest energies (equivalent to a spectral index of α=5/2), and an effective temperature given by 5/9TH, where kTH~180MeV is the Hagedorn temperature measured in collider experiments. Our theoretically obtained formula is in very good agreement with the experimentally measured energy spectrum of primary cosmic rays.