we're literally on the precipice of water wars

Start a car detailing business. Use the water to wash the cars.

Use the money from new, low-overhead business to do anything you want.

Almost no matter how you do it, it's going to be a horrible waste of good drinking water to try to extract cooling from the temperature of the water. If you are in a dry climate, make a DIY swamp cooler. Otherwise shell out for a small AC unit.

Also; using your free lease-included water for stuff like that, is probably the quickest way to no longer have water included in your lease...

If the actual problem is that you yourself are too hot, cool yourself instead. A trick I've picked up working in kitchens, where it's very fucking hot indeed, is to wet your nape and forearms regularly. You can wear a wet hat too. Doesn't really take advantage of the unlimited water but it gets you there.

Use the water pressure to power a turbine generator, then use the electricity from that generator to run a window AC unit

Gut an AC from the dump. Replace the condenser with a tube in tube heat exchanger, using your cold water as a heat sink. Brazed plate HX if you're feeling rich. Replace the cap tube with a TXV for better load tracking. Recharge with R290.

T Sure this is even further beyond your skill level but is the best possible way to use a source of cold to chill your apartment. You can locate it anywhere convenient, not just by the window. You could likely get a COP over 5 and be discharging the water in a fairly modest stream at around 30-40C.

Wouldn't be too difficult to jerryrig a system which does that but because it's going to be a huge waste of water I feel morally obligated to not even give you any ideas. Invest in a split AC system instead. They make ones for windows as well.

Just get a window AC if you have the right type of windows. Otherwise, a dual hose portable heat pump from costco or other reputable source.

A swamp cooler would potentially be a solution; those could be used in conjunction with your ready access to water in order to keep your home cool! Sadly, this is not great for areas of the world which already have high levels of humidity in the air. I don't know if it would work well for you based on not knowing your climate. https://en.wikipedia.org/wiki/Evaporative_cooler

https://what-if.xkcd.com/91/

1. Find a pair of vehicle radiators that are as close to a box fan in size as possible.
2. Zip tie them to either side of the box fan. As the fan blows: it will draw air in through the “second radiator” and blow it out through the “first radiator”.
3. Hook the out of the first radiator to the in of the second using flexible hoses. Cheap garden hoses might even fit.
4. Hook other hoses to the in of the first radiator and the out of the second.
5. Run water on through the first radiator, out of the second. This makes the most efficient heat transfer possible.

Take a cold bath?

Get a kiddie pool, fill it, soak your feet in it

Is there a reason why you can't get an air conditioner? There's tons of valid reasons why it wouldn't be an option, I'm just wondering what your situation is. Because nothing is ever gonna work anywhere close to as good as an air conditioner. If you can afford even the smallest air conditioner then it'll beat every single diy method in most situations

Bottle it, sell it for profit, use the profits to buy a better place to live?

As a kid I used tubes, a box fan, a cooler, and bucket with a siphon to cool me down.

You could easily set that up with just the water from a sink and some hardware store parts.

Search for 'diy fan cooling tub copper coils' as a start.

As an example: Homemade AC - The "Copper Coil" Air Cooler! - (Simple "Box Fan ...

Copper coils have the best thermal efficiency, but plastic tubing would also work.

Depending on you humidity, you could look into building a swamp-cooler. Sounds weird but works pretty great.

Why not buy a portable AC?

Spray water in front of a fan. Way faster than forcing heat exchange through air to metal.

The reason it works on Winter is that the temperature difference is about 50C or more. On a hot 35C day it would mean the radiator is at most at -15C.. And that's why you'll prefer to stand in front of the fridge that has 6C than a radiator that may run at best 15C water inside.

If you have a way of getting unlimited free ice (unlimited free cold water also works), you can run it through a radiator. Get a radiator of any kind, get a water pump and find a way to run the water in the bucket through the radiator. Put the radiator in front of a fan. As long as there is cold water running through the radiator, it will produce at least some cooling. Just don't think you can create ice in the same room you're trying to cool, a fridge generates more heat than the cold it produces. This barely works whenever I have to do it but it is better than nothing and it can make a difference if you just need the room to be a little cooler so you can sleep.

The entire problem with this is that ice or cold water is a really shit form of cooling and the only thing that's any better is.. Compressor a/c. Peltier plates look good on paper but once you hook them up to a heat transfer system, you'd be surprised how useless 500 watts worth of peltiers are for transferring heat. Swamp coolers only work on the desert. Ammonium nitrate can generate cold when it gets dissolved into water but the only way to make a sustained cooling system out of that involves boiling the water to get the ammonium nitrate back so you can reuse it so fuck that.

Alternatively, scientists invented some kind of metal that gets cold when you bend it but good luck figuring out how to make that.

I don't know how big your apartment is but why not a window unit. It's probably the most efficient way to cook your apartment down short of redesigning the building.

Could you get a fountain?
Specially if it spreads the water like rain, it'll help cool the air around, you'd just have to change the water every so often.

If where you live is not very humid you could investigate into swamp coolers.

There are.