The majority of cosmic rays (CRs) generated by star-forming galaxies escape them and enter the intergalactic medium (IGM). Galactic wind termination shocks might also accelerate CRs. I show that the mean pressure of these CRs can reach to within an order of magnitude of the mean Lyman-α forest thermal pressure. At z > ∼ 1, their pressure may have even been dominant. I also demonstrate that, whichever IGM phase the CRs reside in, they contribute significantly to its pressure if its temperature is ∼ 10 4 K, as long as pionic and Coulomb losses are negligible. Where CRs end up depends on the structure and strength of intergalactic magnetic fields. I argue that CRs end up at least 30 kpc from their progenitor galaxies. CRs may self-confine in the IGM to the sound speed, generating > ∼ 10 −13 G magnetic fields. These considerations imply the existence and importance of a nonthermal IGM.