How did dark matter become a relic? 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.
 A: There are two particle candidates for dark matter, and both are still in the realm of hypothesis. These particles are candidates exactly because they cannot decay to something lighter,  they are stable. Neutrinos, which also cannot decay to something lighter, has such a small mass that they cannot model the way dark matter is attracted to gravitational centers.

The axion is a hypothetical elementary particle postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interest as a possible component of cold dark matter.

There is an experiment trying to find the axion as  dark matter .

The Axion Dark Matter Experiment (ADMX, also written as Axion Dark Matter eXperiment in the project's documentation) uses a resonant microwave cavity within a large superconducting magnet to search for cold dark matter axions in the local galactic dark matter halo. Unusually for a dark matter detector, it is not located deep underground. Sited at the Center for Experimental Nuclear Physics and Astrophysics (CENPA) at the University of Washington, ADMX is a large collaborative effort with researchers from universities and laboratories around the world.

Then there is the neutralino, as a candidate for a weakly interacting massive particle (WIMP)

In R-parity conserving models, the lightest neutralino is stable and all supersymmetric cascade-decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum in a detector.

So it is conservation of R parity that makes it stable

Weakly interacting massive particles (WIMPs) are hypothetical particles that are thought to constitute dark matter. There exists no clear definition of a WIMP, but broadly a WIMP is a new elementary particle which interacts via gravity and any other force (or forces), potentially not part of the standard model itself, which is as weak as or weaker than the weak nuclear force, but also non-vanishing in its strength. A WIMP must also have been produced thermally in the early Universe, similarly to the particles of the standard model according to Big Bang cosmology, and usually will constitute cold dark matter. 

The Picasso experiment is looking for dark matter that has spin, which the neutralino does, but the axion does not.
This is an active area of research, both experimentally and theoretically
