Since you are saying with the dark matter freeze-out scenario, the amount of dark matter is now constant
Exactly. It is constant from the time dark matter freezed out.
Is this the same G constant that Einstein used in his theory of a static universe?
The static universe theory has been ruled out for some time (by data). I am however not so sure to get the question. G is the gravitational constant, connected to the force of gravity. It is not a density ad is thus not connected to the density of dark matter today.
I made a mistake... I thought that the Gravitational constant Einstein used had the same value as the energy of dark matter in the universe
No problem. Humans are allowed to make mistakes :p
I suppose that what you mean by 'energy of dark matter' consists of its density :)