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We investigate the temperature distribution of CO-dark molecular hydrogen(H2) in a series of disk galaxies simulated using the AREPO moving-mesh code.In conditions similar to those in the Milky Way, we find that H2 has a flattemperature distribution ranging from 10 - 100 K. At $T < 30$ K the gas isalmost fully molecular and has a high CO content, whereas at $T > 30$ K, the H2fraction spans a broader range and the CO content is small, allowing us toclassify gas in these two regimes as CO-bright and CO-dark, respectively. Themean sound speed in the CO-dark H2 is 0.64 km/s, significantly lower than thevalue in the cold atomic gas (1.15 km/s), implying that the CO-dark molecularphase is more susceptible to turbulent compression and gravitational collapsethan its atomic counterpart. We further show that the temperature of theCO-dark H2 is highly sensitive to the strength of the interstellar radiationfield, but that conditions in the CO-bright H2 remain largely unchanged.Finally, we examine the usefulness of the [CII] and [OI] fine structure linesas tracers of the CO-dark gas. We show that in Milky Way-like conditions,diffuse [CII] emission from this gas should be detectable. However, it is aproblematic tracer of this gas, as there is only a weak correlation between thebrightness of the emission and the H2 surface density. The situation is evenworse for the [OI] line, which shows no correlation with the H2 surfacedensity.

CO-dark gas and molecular filaments in Milky Way-type galaxies - II: The temperature distribution of the gas