Ah, I think I can help with this one... Break down the characters into "隔水" and "加熱"/ "冷卻". You now have effectively a "partition of water" and "heat up"/ "cool down". So... 1) 隔水加熱= You can say, "double-boil", but that is because there no exact translation for this cooking method in Western style cooking. This is often used in Asian cooking recipes, and just means to put a pot of food into a vessel, and place it into another vessel full of water to stew the food. In western style cooking, a similiar cooking method uses a bain marie (or a water bath). So yes, go ahead and use "double boiling" if you want to specifically talk about this method of Asian cooking. (You may get quizzical looks, so be prepared to explain). If not, then it is to use a "bain marie" or "water bath". 2) 隔水冷卻= This is yet another method of cooking, but specifically for cooling down after you have heated the pot of food in the bain marie. Again, this is unique to Asian style cooking. However, in English just say "cooled rapidly under cold water" or "placed in an ice bath". Similar methods in Western style cooking places a high pressure pot under water to rapidly cool down the pot prior to opening it, or to rapidly cool blanched vegetables under an ice bath. What an interesting question~

I'm confused. Are you putting the pot in hot water? Why? Or are you putting hot water into a pot and then putting something else into the pot with the hot water? If it is something else - what are you putting in hot water (or cold water) and for how long is it there? And why? Is it food?

"Double-boiling" and "double-cooling" aren't terms that are used in American English. If we're heating water to cook, we just call it "boiling". If we need to put something in cold water, we just say "soak in cold water"