Uhm, I may not have any -advanced- knowledge, but at school they taught us to cool burns with cool water for a while, but never to put ice on them, because then the very low temperature might add cold damage to the burnt area easily. Something like that. After the fact like that, I think it's more appropriate to treat the wound than try to cool it.
Honestly I don't buy that heat from the burn itself would remain for so long. Blood circulation should take care of that. If the spot is warm, it's probably because there is a lot of blood circulating there to heal the broken tissue, and that is probably a very good thing, even though it hurts.
I was unsure, so I looked it up, and I found this. No rocket science and no doctors involved, but honestly this makes perfect sense to me. Cooling the burn down is only relevant when the heat from the burn's source is still there and might continue burning additional tissue.
http://answers.yahoo.com/question/index?qid=20070627182314AAClUfJAnyway, Alexia, don't worry about it too much for now. I'm not sure I'd even go to the hospital unless it was visibly getting problematic. Well, it depends on your options. If you can see a nurse, doctor, etc. relatively easily, well by all means do so. They may have tips to maximize healing and minimize eventual scarring.
For home treatment, I'd cover it with a thick layer of aloe gel, or whatever treatment you use on burns and sunburns, repeatedly (basically don't let it dry), make sure it doesn't get infected and wait a little. The key is to keep the skin out of the air and contact in general to minimize pain (contrary to what so many moms will tell you, putting a wound in the air does not help its healing process at all).
This guide just uses bandages, but honestly, I can't imagine that not being really painful, putting a dry bandage on a burn like that. I've always kept burns out of air with aloe gel, because dunno, there's no friction, and to me it makes sense to help the skin heal and moisturize it with aloe, so that the salvageable zone has good chances of surviving. And let's be honest, a burn just feels so much better when it's kept wet.
Oh, and make sure what you put on it won't damage your skin further. They sell aloe gels that also contain alcohol (diffuses the fragrance they put in it, feels cooler because it evaporates along with some heat, but dries the skin, which you don't want, especially on burned skin) and/or menthol (cooling sensation without any benefit, and I think it has bad effect on skin, too, maybe drying, but I don't remember). Of course, the amounts they add shouldn't be a big deal, and it would be surprising if the bad effects of alcohol outplayed the good ones from the aloe. But you don't want to take a chance, do you?
By the way, normally, aloe gel is clear. I was shocked to learn this. The aloe extract they use in products doesn't have any plant pigment left in it. If your aloe gel is green, it's just that they put colour in it, so that it says "hey, plant juice". Not that colour is so terrible that you should run away from it, but definitely don't distrust a product because it's clear. It's actually probably better if it is.
From your description, the burns sound like they're relatively minor, but I can't know that for sure. Try to determine your kind of burn and react appropriately.
http://www.mayoclinic.com/health/first-aid-burns/FA00022It should start getting better gradually pretty quickly. If there's signs of worsening, see a health professional if you can.
