Dark matter is not the only possible source of heat in ordinary matter: cosmic rays and similar would also heat ordinary matter. Experiments searching for dark matter see a great deal of heat from cosmic rays and look very hard for but have not yet found dark matter, which is looked for primarily by the heat it deposits. This is to say: when dark matter hits a nucleus the nucleus recoils, depositing some energy in a detector, but causing very little ionization, relative to (most) cosmic rays. This energy deposition quickly (particularly in CDMS, but also in other experiments) becomes heat which (in turn) is detected directly because it heats a bolometer or indirectly because it (for example) nucleates bubbles. With careful experimental techniques that allow the energy deposited to be seen quickly and distinguished from other energy depositions. These experiments show that there is orders of magnitude more heating / deposition of energy from cosmic rays than from dark matter, and by extension this is true for all matter not well shielded from cosmic rays e.g. effectively all matter we can imagine "seeing". Actually, this is too weak a statement: even in well shielded locations (deep mines) there is much more heat deposition from cosmic rays than dark matter. So, (in effect) I think that the best recent published limit on dark matter detection will for the forseeable future be the best limit on heating from dark matter. I suppose, this assumes that we know pretty well what the relative cross section of dark matter with different kinds of matter is. I suppose that if, contrary to all expectations, dark matter interacts strongly with something not yet used in a detector and weakly with stuff that has, this could be wrong. But, that is "not expected".
Depending on what quanta dark matter truly is made of, the Higgs boson may or may not be able to decay into those quanta. Being further nit-picky, at colliders, where the Higgs boson can be studied, dark matter may or may not be produced, the collider experiment will never be able to tell whether a new particle has a long enough lifetime to be the cosmological dark matter. One thus typically refers to such decays as "Higgs to invisible". This avoids the reference to dark matter and would be the right buzz words to use in an internet search.
Yes, there are many dark matter candidates that the Higgs could decay into, yes, experiments at colliders could be able to detect such events, but current searches have not come up with significant discrepancies.
This paper from the CMS collaboration is a nice example to illustrate these points. Their Figure 1 shows some example Feynman diagrams where the Higgs ("H") decays into invisible particles (invisible in CMS, labeled $\chi$) that might be dark matter:
They go on and search their data for such events. Figure 6 from that paper shows what they find:
The colorful areas in the top portion of that figure are various standard model processes that are expected to contribute to this analysis channel. Black dots are their data, which agrees with the sum of the standard model processes. This is again shown in the middle portion where the ratio of the data to the standard model expectation is everywhere about one. The dashed line is the strongest possible amount of decays that they could attribute to the standard model Higgs to decay to some invisible particles.
This then allows them to calculate the largest possible branching ration of Higgs decaying into new beyond-standard model invisible particles, relative to the decay into standard model invisible particles, as being less than 20%, as they show in their Figure 9:
Best Answer
The original experiment was designed to find it as a proof of antimatter, not dark matter.
And seem to be holding up for that, leftover antimatter from the big bang.
You ask:
This is a competing theoretical explanation, based on the theoretical model, example
It would be one with very small probabilities because It would have to be a complicated single interaction with 12 quarks and 12 antiquarks, if it were a pair production from a very high energy gamma-field pair production. The models where anti helium coalesces from lower antibaryon number have better probabilities.
Anti-deuterons also may appear by coalescence but there are proposals that if dark matter is composed of supersymmetric particles :
It is all very speculative and theoretical.