Why did the decay rate of the dark matter particles fall when the temperature of the Universe $T_U$ dropped below dark matter mass $M_{DM}$?

In particular, why can it not decay into lighter particles even today? After all, from zero temperature field theory, I have the experience that heavier particles can decay into lighter ones even at zero temperature. However, I'm not familiar with finite temperature calculation of decay rates. Does the decay rate (as calculated in finite temperature field theory) go to zero as the temperature falls below a certain threshold?

I understand that the production of dark matter would stop below a certain temperature but I don't understand why should the decay stop.