According to various sources, the human heart pumps 5 liters of blood/minute at rest and 15-20 liters of blood/minute during running or exercise, 6,000 to 7,500 liters of blood/day (which is equal to 4⅙ to 5 5/24 liters of blood/minute), 4.73 to 6.62 liters of blood/minute, and so on and so forth. As such, 5 liters of blood/minute seems a reasonable placeholder value in regards to flow rate through the heart for the purposes of this question.
According to various sources, the specific heat capacity of blood is 3.49, 3.5, or 3.617 kilojoules per kilogram per degree Celsius increase in temperature. I'll use 3.5 KJ/KG/C for the purposes of this question.
There are more sources than I can list in regards to the density of blood, but most agree that it's roughly 1,025 to 1,060 kilograms per cubic meter.
5 liters blood pumped/minute = 0.005 m^3 blood pumped/minute. 0.005 m^3 blood pumped/minute * 1,060 KG/m^3 blood density = 5.3 KG blood pumped/minute. 5.3 KG blood pumped/minute * 3.5 KJ/KG/C = 18.55 KJ required to cool 5.3 KG blood by 1 degree C.
According to chemistry-reference.com, there's a chemical reaction that uses up a mole of ammonium chloride/sal ammoniac, a mole of lye/sodium hydroxide, and 30.56 kilojoules of energy to produce a mole of water, a mole of salt/sodium chloride, and a mole of ammonia. In other words, I only need a single mole each of lye and ammonium chloride to cool more than a minute's worth of pumped blood by 1 degree Celsius - as a matter of fact, we can probably cool the blood as much as we want, since the reaction really doesn't need much of this stuff in order to function - the primary problem is venting the ammonia waste product.
A creature I'm working on takes advantage of this. Right before its blood vessels feed into its heart, a web of them - a rete mirabile, or, in technical terms, a cooling jacket - wraps around a biological chemical reactor within the creature's body, which is where this reaction occurs. When physically exerting itself, it triggers a valve in that chamber (a la a bombardier beetle's chemical-mixing mechanism), carries out the above chemical reaction, vents the resultant ammonia - which it can regenerate later - to the atmosphere, stores the salt in some kind of special receptacle, and pees the water out later. In exchange, the creature's blood gets heat sucked out of it, thereby reducing the rate at which it overheats.
The question: how cold can this creature make its own blood before said blood stops working? After all, as I said above, the blood can be cooled as much as I want; the unknown variable here is how much it can be cooled before bad stuff starts happening.
I recognize that the viscosity of blood increases as its temperature decreases, meaning that low-temperature blood is harder to pump, but that's not really the problem here since, after getting the heat pulled out of it, the blood will rapidly begin picking up more heat and its temperature will increase again. However, is there a temperature at which red blood cells can no longer carry oxygen, or at which they freeze so much that they can't function again - i.e. where it doesn't matter how much they get re-heated, because the sudden drop in temperature irreparably destroyed them? I'm currently thinking that that's somewhere between "freezing" and "normal body temperatures".
Don't worry about anything else related to this creature's rather unique biology - i.e. how it produces/stores ammonium chloride and lye or vents ammonia. That's not what this question is about. Moreover, this creature is, for all other aspects - including the makeup of and chemical structure of its blood - a relatively normal human being.