We present new far-ultraviolet observations of the young M8 brown dwarf 2MASS J12073346-3932539, which is surrounded by an accretion disk.The data were obtained using the Hubble Space T elescope-Cosmic Origins Spectrograph. Moderate resolution spectra (R ≈ 17,000 -18,000) obtained in the 1150 -1750 Å and 2770 -2830 Å bandpasses reveal H 2 emission excited by H I Lyα photons, several ionization states of carbon (C I -C IV), and hot gas emission lines of He II and N V (T ≈ 10 4 -10 5 K). Emission from some species that would be found in a typical thermal plasma at this temperature (Si II, Si III, Si IV, and Mg II) are not detected. The non-detections indicate that these refractory elements are depleted into grains, and that accretion shocks dominate the production of the hot gas observed on 2MASS J12073346-3932539. We use the observed C IV luminosity to constrain the mass accretion rate in this system. We use the kinematically broadened H 2 profile to confirm that the majority of the molecular emission arises in the disk, measure the radius of the inner hole of the disk (R hole ≈ 3R * ), and constrain the physical conditions of the warm molecular phase of the disk (T (H 2 ) ≈ 2500 -4000 K). A second, most likely unresolved H 2 component is identified. This feature is either near the stellar surface in the region of the accretion shock or in a molecular outflow, although the possibility that this Jovian-like emission arises on the day-side disk of a 6 M J companion (2M1207b) cannot be conclusively ruled out. In general, we find that this young brown dwarf disk system is a low-mass analog to classical T Tauri stars that are observed to produce H 2 emission from a warm layer in their disks, such as the well studied TW Hya and DF Tau systems.