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We report the detection of the CO(12–11) line emission toward G09-83808 (or H-ATLAS J090045.4+004125), a strongly-lensed submillimeter galaxy at z = 6.02, with Atacama Large Millimeter/submillimeter Array observations. Combining previously detected [O iii] 88 μm, [N ii] 205 μm, and dust continuum at 0.6 mm and 1.5 mm, we investigate the physical properties of the multi-phase interstellar medium in G09-83808. A source-plane reconstruction reveals that the region of the CO(12–11) emission is compact ($R_\mathrm{{e, CO}}=0.49^{+0.29}_{-0.19}\:\mbox{kpc}$) and roughly coincides with that of the dust continuum. Non-local thermodynamic equilibrium radiative transfer modeling of CO spectral-line energy distribution reveals that most of the CO(12–11) emission comes from a warm (kinetic temperature of Tkin = 320 ± 170 K) and dense [log (nH2/cm−3) = 5.4 ± 0.6] gas, indicating that the warm and dense molecular gas is concentrated in the central 0.5 kpc region. The luminosity ratio in G09-83808 is estimated to be LCO(12-11)/LCO(6-5) = 1.1 ± 0.2. The high ratio is consistent with those in local active galactic nuclei (AGNs) and 6 < z < 7 quasars, the fact of which implies that G09-83808 would be a good target to explore dust-obscured AGNs in the epoch of reionization. In the reconstructed [O iii] 88 μm and [N ii] 205 μm cubes, we also find that a monotonic velocity gradientmore »is extending over the central starburst region by a factor of 2 and that star-forming sub-components exist. High-resolution observations of bright [C ii] 158 μm line emissions will enable us to characterize the kinematics of a possible rotating disk and the nature of the sub-components.