Near boiling water is about 4% less dense than near freezing water. If you filled the buckets right to the top, there would be less water in the hot bucket.
This may happen but, in a normal freezer, the action of the compressor is meaningless.
My feezer is 70% full, with probably 30kg of stuff in it. This stuff is mostly at -22C (I have a thermometer), and so, this mass is what cools new things added to the freezer, fast.
(Why -22C instead of -18C? Salt water fish is in there, and I also want a little headroom.)
For example, most manuals urge one to not have an empty freezer. That cold mass is one reason why.
When I was a kid, we had a fridge from the 50s in a cottage. When I started drinking, we'd put 5 cases of beer (24*5) in glass bottles in there, and 2 hours, yes hours later they'd be at 1C.
Now, modern fridges/freezers literally do not have that degree of cooling power. This is on purpose, for it is more efficient to power a small compressor all the time, than a massive compressor for 10 minutes.
You may want to argue this point, and that's fine, but I am merely providing info both from the manuals of modern freezers and fridges, which I have read, and from online when looking at why they are so bloody slow to cool things.
My LG fridge manual actually says not to put warm meat in my freezer, unless the thing has loads of frozen stuff in it.
Otherwise the meat could go bad before freezing.
Oh, another wonder of modern fridges. If you buy one and put it outside, or in an unheated garage, the freezer becomes useless.
This is because many fridges have no thermometer in the freezer part, and only get a reading when the compressor comes on, to cool the fridge.
As the fridge is always cool when it is 3C outside, or cooler, the freezer never keeps stuff frozen.
> My LG fridge manual actually says not to put warm meat in my freezer, unless the thing has loads of frozen stuff in it.
That sounds silly to me. In my experience ice cubes freeze within an hour and meat within a couple hours.
I don't know if I'd call it trivial, bringing 1 kg of water from 85C to 0C involves about 100W for a duration of one hour whereas a typical freezer removes heat at, what, 200W or 300W?
That looks like the amount of energy needed to heat that water.
But cooling things doesn't work the same way. Instead you want to extract that energy/move it somewhere else. The efficiency of some approaches, unlike when heating something, can hugely depend on ambient temperature. It's possible you'll be able to extract more useful energy cooling something than you'll have to expend!
Introducing hot water into a freezer rapidly raises the ambient temperature of the freezer. This causes the thermostat to click on, and the heat pump therefore runs until the ambient air drops down below the thermostat's shut off point. A lot of heat is removed very early on, and the ambient freezer temp is at its lowest going into the liquid to solid phase change.
With the cold water, the ambient air of the freezer doesn't rise nearly as much. The thermostat does n't click on until a long while later, once enough heat leaks out of the freezer. The ambient air is warmer going into the phase change.
It doesn't work if you put the two cups in the same freezer, but at least it could explain some anecdotes.