I keep approximately 5 minutes of temp readings (used a TMP36 sensor) at 1 minute intervals using an RTC (Real Time Clock) to more accurately time it. I made a small 'power tester' circuit that basically acts like a normally open button and closes when there's no power coming into the freezer. I may or may not have spliced into the wiring of the freezer for this and lied to my wife about it when she asked if I was going to burn the house down.
Side note: Is it just me or does it seem like you get asked this question frequently?
I have the 'button' circuit wired into an Arduino I/O pin just like you would an old normally open button. When the 'button' circuit closes the Arduino triggers a VERY audible siren I cut out of an old water leak detector. If I have 5 minutes above my temp threshold the siren goes off. I upgraded it over time to use a small LiOn battery and got a circuit to switch to it when incoming power is cut for whatever reason, and basically when in battery mode it's also siren time. I got through 10 minutes of siren battery testing before deciding it was good enough and I was tired of hearing it.
We've talked about doing something similar for the propane fridge and an activity sensor in my friend's cabin in remote central Oregon using a 3G breakout for communications but we've run into the issue that the 3G there is really spotty and we just couldn't reliably get a data connection out there. Also debugging electronics with no internet, running off generator and batteries, and in the dark because we were fishing all day doesn't lend itself to really getting the job done.
Since you have WiFi you may have what you need to do it more simply, although the 3G might be a safety net for the safety net once you get your solution done. My biggest problem at the cabin, besides the lack of cell coverage, is the lack of constant power. There's barely any solar power, and it's used to keep the batteries conditioned, and it's often weeks or months between people getting out there. I would think at your place you could pretty much just use my approach and add WiFi, or just use an ESP8266 instead of an Arduino and do everything else the same. The battery switchover stuff was the biggest pain in my butt, mostly just because I've not got a ton of LiOn experience.
EDIT: On the RTC and temp logging, it's really not necessary. I just did it to do it. You can use the oscillation ticks and a counter to get close enough. Hell, you can just bump the threshold up a little and raise the alarm as soon as you see it without logging anything and get a drastically simpler circuit.