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 Telescope-Cosmic Origins Spectrograph. Moderate resolution spectra (R~17,000-18,000) obtained in the 1150-1750 A and 2770-2830 A bandpasses reveal H2 emission excited by HI Ly$α$ photons, several ionization states of carbon (CI - CIV), and hot gas emission lines of HeII and NV (T ~ 10^4-5 K). Emission from some species that would be found in a typical thermal plasma at this temperature (SiII, SiIII, SiIV, and MgII) 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 CIV luminosity to constrain the mass accretion rate in this system. We use the kinematically broadened H2 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(H2)~2500-4000 K). A second, most likely unresolved H2 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 H2 emission from a warm layer in their disks, such as the well studied TW Hya and DF Tau systems. Comment: ApJ, accepted. 12 pages, 10 figures. 3 tables.