Folks reading headlines about record warm oceans might be surprised by this question. But it's a real question, if perhaps from a different viewpoint than you might think.
If we look at the surface of the ocean, we see that most of the ocean is warm. The presentation at the link over-emphasizes the polar regions -- they're actually much less of the earth's area. Even so, over half the ocean -- surface -- is warmer than 20 C.
So you might figure that the volume of the ocean would also be some moderately warm figure, maybe a bit colder since cold water sinks, but still fairly warm. Surely over 10? In fact, the volume of the ocean -- average up every blob of water there is -- averages 3.5 C. Go back to the surface map and take a look at how much of the ocean is that cold. Answer: Not much. Even less when you allow for the fact that the map is exaggerating how big the polar regions are (I get about 14% of the ocean surface was at least that cold on February 26th). The importance of those small areas of cold water is that there is no refrigerator in the ocean. Once water leaves the surface of the ocean (except for one even smaller exception I'll get to), there's no way to make the water any colder. So if you see water that's -1.0 C in the ocean, you know it came from somewhere that had water at least as cold as -1.0 C. It could have been even colder -- the original cold blob might have mixed with a warmer blob of water.
Some of you might have objected up there when I mentioned that the average ocean temperature is 3.5. There's a fairly popular error that says the deep ocean has to be 4 C. It runs this way: Water is densest at 4 C, so as you cool a body of water, once it reaches 4 C all this cold water sinks to the deeps. As you cool the surface further, the water is less dense, so it quits sinking. That leaves you with 4 C water in the deep.
The thing that is wrong with that, and it's only one error, is that the statements are only true for fresh water. If you're looking at ocean water -- where the density and freezing point also depend on how much salt you have, things are different. For saltwater with salinity greater than 22, the water gets denser all the way down to the freezing point. Typical ocean water has a salinity of about 35, and extremely little of the ocean is as fresh as 22. So you can get as cold as about -1.9 C in the ocean before freezing, and water as cold as that sinks to the bottom. (Or at least it tries.)
One part of our story, then, is that the deep ocean is cold, very cold, because cold water is denser than warm water and salt water gets denser all the way to its freezing point. We can look at the maps of surface temperature and see that the only areas that get cold enough to supply most of the volume of the ocean are the Antarctic, the North Pacific and Sea of Okhotsk, the Labrador Sea (east coast of Canada), and the Nordic Seas (between Greenland, Iceland, and Norway). If we look at the bottom of the ocean, we see even colder temperatures than the 3.5 C average. Makes sense -- the bottom would have the coldest, densest water of all. It's down to 2 C or even -1. That limits the possible source regions even farther. We find that the bottom waters (our creative name for the waters on the bottom of the ocean) come from either the Antarctic (Antarctic Bottom Water) or North Atlantic (the Nordic Seas, supplying North Atlantic Deep Water) -- and not the Pacific.
That points us to a different issue. Temperature is not the only thing that affects the density of water. How much salt is present does as well. The North Pacific is fresher than the North Atlantic and Antarctic. So even if we cool it to the freezing point, the North Pacific water can't get dense enough to sink to the ocean bottom.
But ... is there a place that has really salty water? Can it get salty enough to make up for being warmer? The answer is yes, and yes/no. The yes is, predominantly, the Mediterranean Sea. Water flowing out from the Mediterranean sea has a salinity of about 38, versus the about 35 of average ocean water. That makes it quite dense.
The yes/no is a matter of history. The answer today (and for a long time) is no. Which is a little surprising. The thing is, if you took the temperature and salinity of Mediterranean Outflow Water, and of Antarctic Bottom Water, and computed their density at the surface of the ocean, the Mediterranean water is denser. So you'd expect it to be in the bottom of the ocean. But, if you compute the density at depth -- where the water would be trying to get to if it wants to be bottom water -- you find that below a certain depth, the Antarctic water is denser. Because Antarctic (and North Atlantic) water does fill the ocean below that depth, the Mediterranean water can't sink through to the bottom. But if Mediterranean water were already filling the deep ocean, then it is the Antarctic and North Atlantic water that would be blocked from reaching the deep ocean. An ocean structured like this would be extremely different from the one we do have today.
Let's go back to the one tiny exception to my 'no subsurface refrigerators' rule above. It involves another subtlety in how water behaves. Namely, the freezing point of water, like all materials, depends on the pressure. The higher the pressure, the lower the freezing point. Now think about the Antarctic Ice Shelves, the big ones -- the Filchner-Ronne in the Weddell Sea, and the Ross Ice Shelf in the Ross Sea. They each have an area of about 500,000 km^2 (200,000 square miles), about the size of France or Texas. Important to our concern here, they are very thick, from 300 m at the front edge, up to 1000 meters back where they meet up with the Antarctic ice sheet. Water that was near the freezing point -- at the surface -- can circulate under these ice shelves and cool even farther, to the freezing point of water under pressure. It's a small effect, about 0.075 C per 100 meters of ice shelf thickness. But it isn't zero, and turns out to be important for the 'Ice Shelf Water' method of making Antarctic Bottom Water. So you can see water at -2.3 C thanks to this particular refrigerator. (We talk about water mass 'formation'; it's really a matter of figuring out how things get cooled, saltier, or mixed together.) Still, at a total of 1 million km^2, versus 360 million km^2 of the ocean, it's a pretty small exception to the rule.