An aftermarket cooler would dump heat from the CPU to the room faster, so it would raise the temperature of the room faster(not more, just faster)
A bad cooler would take more time to dump the same amount of heat, and since your room would be losing heat(assuming its cold outside) at a certain rate, the overall temp of the room would rise slower.
If I was to run a load for 1 min, the aftermarket cooler would heat the room more(at the time I switch off the load), because the stock cooler(and the CPU) is still holding some heat and is yet to dump it into the room. But if I was to run this experiment over long period of time, both the heatsinks would reach their peak dissipation rate, and the difference would be negligible. But once the CPU is at idle, the aftermarket heatsink would stop dissipating heat, but the stock heat sink would still be dumping the heat it is holding.
So even though the amount of heat dumped in the room is same, the peak temp in the room would be higher with the aftermarket cooler.
There are a LOT more factors that in play here like the heat dissipation rate of the room will increase as the delta of temp between room and outside increases and the rate of dissipation of heatsink varies based on the temperature of the air around it etc