Let's consider a warming of 1 K for simplicity. The central question for the oceanographers is then how deep a layer of the ocean gets warmed. This is a difficult question. The challenge lies in the fact that the atmosphere heats the ocean at the top. Not obvious is that this impedes warming much of the ocean.
Warm water is less dense, so tends to stay at the surface of the ocean.
If this were all that happened, only the layer of ocean directly warmed by the sun would be affected, about the top 100 meters. There is mixing within the ocean, which tends to force some of this heat further down. So mixing brings down warm water, and upwelling brings up colder water. Let's assume that the thickness that gets warmed is approximately the same as that which is already warm. That is approximately 500 meters.