  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 flat temperature distribution ranging from 10 - 100 K. At $T < 30$ K the gas is almost fully molecular and has a high CO content, whereas at $T > 30$ K, the H2 fraction spans a broader range and the CO content is small, allowing us to classify gas in these two regimes as CO-bright and CO-dark, respectively. The mean sound speed in the CO-dark H2 is 0.64 km/s, significantly lower than the value in the cold atomic gas (1.15 km/s), implying that the CO-dark molecular phase is more susceptible to turbulent compression and gravitational collapse than its atomic counterpart. We further show that the temperature of the CO-dark H2 is highly sensitive to the strength of the interstellar radiation field, but that conditions in the CO-bright H2 remain largely unchanged. Finally, we examine the usefulness of the [CII] and [OI] fine structure lines as 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 a problematic tracer of this gas, as there is only a weak correlation between the brightness of the emission and the H2 surface density. The situation is even worse for the [OI] line, which shows no correlation with the H2 surface density. 
CO-dark gas and molecular filaments in Milky Way-type galaxies - II: The   temperature distribution of the gas