After reevaluating the creature that cools its blood with lye and sal ammoniac, I removed the middleman - the cooling chamber and use of blood as coolant - and replaced the potassium hydroxide/ammonium nitrate reaction with urea (that "some of the stuff in pee" the title mentioned) dissolving in water. After all, why transport the heat being removed to a central location when it can be removed on-site instead? Each individual cell in the creature's body now has its own, miniature cooling system, connected to its own system of thermoreceptors. This has several advantages:
Each individual cell can "decide" when to cool itself (whenever its thermoreceptors sense a high temperature and tell their cell's cooling organelles to mix urea and water) as opposed to the hypothalamus deciding when to activate the centralized, body-wide cooling system.
No heat seeps back into the body en-route to the cooling chamber; it gets removed on-site, ensuring as much as possible of it is taken from the cells.
Although the dissolution of urea in water is a weaker cooling mechanism, removing 15.39 kilojoules of energy (see the first page's bottom right) per 1 mole of each substance involved instead of the 30.56 kilojoules removed by potassium hydroxide and ammonium chloride reacting, such a reaction produces ~63.5 milliliters of aqueous solution as its product, rather than ~45 milliliters of salt water...and more than 22.414 liters of toxic, reactive, caustic ammonia gas (even at STP, it takes up 22.414 liters, and the higher temperatures in a living thing will increase that). In other words, while urea dissolving in water might be ~1.986 times weaker as a cooling mechanism than potassium hydroxide reacting with ammonium nitrate, its byproducts take up more than 353 times less volume.
Urea is already produced by Earthly biology on a regular basis, and is therefore much more readily available than lye or sal ammoniac.
The body has about ~37.2 trillion cells. Taking a fairly average 70-liter person and dividing their volume by 37.2 trillion finds the average volume of each of their cells: ~1881 cubic micrometers. If a mere 1% of that volume is reserved for reactants and 4% of that volume is dedicated to extra thermoreceptors and the organelles for containing and reacting the reactants, that's 18.81 cubic micrometers for reactant storage per cell, or 699.732 milliliters/cubic centimeters across all 37.2 trillion cells.
Urea has a density and molecular mass of 1.335 grams per cubic centimeter and 60.056 daltons/atomic mass units per molecule, respectively; since the value of the molecular mass of a molecule in daltons is the same as the molar mass of said molecule in grams, that gives urea a molar mass of 60.056 grams. 60.056 grams/1.335 grams gets us a molar volume of urea of ~44.986 cubic centimeters per mole. Water has a density and molecular mass of 0.9970 grams per cubic centimeter and 18.015 daltons/AMUs per molecule, respectively, giving it a molar volume of ~18.069 cubic centimeters per mole.
44.986 + 18.069 = 63.055. Since 699.732 cubic centimeters are on hand to store urea and water, that means we can store ~11.097 moles of each. Dissolving 11.097 moles of urea in 11.097 moles of water uses up ~170.783 kilojoules of energy, enough to negate the heat from ~19 to ~57 minutes of exercise by a fit adult. I'd say that's a fairly good amount of cooling.
The question, however, is how the urea gets there in the first place. Normally, the body wants to pee urea out; indeed, after each cell in this organism mixes its urea with water, that's exactly what happens: it gets pumped out of the cell as waste. As far as I know, there are no pathways - metabolic, chemical, biological, or otherwise - for a living organism with Earthly biology to deliberately supply what is ordinarily a waste product to its own cells.
As such, the question: via what chemical or biological pathway can a creature with Earth-like biology get a small quantity of urea to each and every one of its cells so that they can have their own cooling system? While this might seem rather in-depth for any piece of writing (or it might not), I'd like to know whether there are any interesting side-effects to this that I could spin out into further wacky biology, so please be as specific as possible, and cite actual chemistry if you can. It'd be fascinating if the chemical transport mechanism to get urea to every single cell could also somehow be used to transport other substances as well.