In most dark matter models, there exists a (non-gravitational) connection between dark matter and normal matter. This allows two phenomena:
- dark matter pair production from the annihilation of two standard model particles
- the pair production of two Standard Model particles from the annihilation of two dark matter particles.
The two processes are actually the inverse of each other and both are playing a big role to explain what is going on in the early days of the universe.
Today, the universe has expanded and cooled down. Therefore, dark matter is too diluted for the process #2 to happen and the Standard Model particles are not energetic enough to produce dark matter (and the process #1 is switched of too).
In the model discussed in this post, dark matter capture by Earth offers a chance to the second process to restart (we have enough dark matter in a small volume so that they can meet and annihilate again). This generates a flux of Standard Model particles that contribute to the heat flow of the planet.
On different grounds, the heat flow of the planet is very well measured: 47 TW (at the 5% level). This means, dark matter could contribute to these 47 TW (assuming that the details of the dark matter model are such that all of what I detailed in the post could happen; this is indeed a model dependent thing). In both cases (dark matter contributions to the 47 TW or not), the 47 TW are still 47 TW. Therefore, if it has an impact on the temperature changes, this impact is already accounted for by the climate models (who do not care about how the 47 TW arise).
Does it clarify? Sorry for the very long answer ;)
It does a little bit. Are we assuming the annihilation of the two particles is spontaneous?
Yes, this is definitely prompt.