Using a cosmological simulation at redshift 5, we find that the baryon-rich cores of intergalactic filaments extending outward from galaxies commonly form isothermal gas cylinders in regions favorable to their formation. The central gas density is typically about 500 times the cosmic mean total density, and the temperature is typically 1-2 times 10^4 K, just above the Lyman alpha cooling floor. These findings argue that the hydrodynamic properties of the gas are more important than the dark matter in determining this structure. It is noteworthy that the temperature and ionization state of the gas completely determine a finite total mass per unit length of an isothermal cylinder. Our findings may have implications for understanding the "cold mode" mechanism of gas transport into galaxies.