Ice evaporates, how do you tell the difference between evaporation and power outage? Sure there should be more loss in the case of an outage but you don't know the exact evaporation rate to expect.
Your computer has a setting on whether or not it should turn back on after power loss, build a low cost circuit that does the same(turns off in case of power loss) and you are done.
Place something like a coin on top of the ice. If the ice melts, the coin will submerge and be inside the ice if it re-freezes. (Assuming you put the ice in a container).
.. and the nice thing is that it's a close representative of what you're trying to measure. If it's warm enough to melt a significant portion of ice, it's warm enough to spoil your food.
I was thinking of some sort of canary that would trip if power turned off, but that would only tell you if the power ever went off, not how long it was off. No sense wasting a fridge full of meat if the power was off for a few seconds.
A clock that requires electricity but displays the time mechanically with a date on it would also suffice, as you could measure the delta between what it displays and what the time really is (accounting for normal inaccuracies).
A weakness of the clock idea is that the power could be off multiple short periods; too short to allow the meat to spoil, but enough in sum to make the clock pretty far off.
Battery-backed microcontroller logging all power off events and temperature changes :)
If the battery has run out, then there has been enough long power outages that the meat is probably spoiled. If the battery hasn't run out, you have a nice log of what happened when.
Not that I think its better than the ice solution, but theres certainly techie options that you could build for $10 or so.
Your computer has a setting on whether or not it should turn back on after power loss, build a low cost circuit that does the same(turns off in case of power loss) and you are done.