Hot dark matter would be made from very light, fast moving particles. Such particles could not possibly be gravitationally bound to any structure, but rather would be dispersed all across the universe.
But dark matter is always "found" (or "inferred") either gravitationally bound to some visible structure (e.g. weak lensing detection of dark matter associated with colliding galaxy clusters / flat rotation curves of spiral galaxies / abnormal velocity dispersion in galaxy clusters) or not associated to anything visible but nevertheless forming clumps (weak lensing detection of galaxy clusters previously unseen). That is why dark matter is thought to be cold.
Additionally, there is a clear distinction between both types: there is not such thing as dark matter that is "not too cold but not too hot either" (see footnote as well). Dark matter is either made of particles with less than ~10 eV (hot dark matter, made of light particles, mostly dispersed everywhere) or particles with more than ~2 GeV (heavier, slower particles gravitationally bound to some structure). Both limits are found when imposing the maximum amount in which the candidate particles (neutrinos or something more exotic) can possibly contribute to the actual value of the density parameter due to matter in our expaning Universe.
Thus, either DM appears gravitationally bound (cold DM) or dispersed (hot DM), and both types are clearly distinct (10 ev vs 2 Gev). Observations favour the first case. However, Cold Dark Matter is not the ultimate solution, and still faces some problems.
Regarding the possibility of mixed solutions, many of them have been already ruled out. Microlensing has ruled out the possibility of unseen compact objects (brown dwarfs, stars, stellar black holes) in galactic halos, in our galactic neighbourhood as well as in the extragalactic domain. Ordinary matter (stones, bricks, dust) cannot possibly be, otherwise they would become hot and re-radiate. Any exotic mix of known particles doesn't work.
All we think we know is that DM must be made of some heavy particles yet to be discovered. In order to introduce a more complex model (e.g. different types of particles depending on the structure they appear attached to) one needs a justification (i.e. some predictions that better agree with reality) and nobody has been able to do that yet.
Remark Note that Dark Matter particles, either from the hot or the cold type, cannot possibly "slow down" and clump too much (e.g. forming planets) because they don't interact electromagnetically like ordinary matter, that is why DM is said to be collisionless. Wherever infalling ordinary matter forms any structure (e.g. protostars or accretion disks), a very important part of the process is thermalisation, i.e. the redistribution of energy of the infalling particles by means of numerous collisions. This cannot happen with Dark Matter.
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