BeagleBone Black specs | Just Board 

Thanks! I've got more of those planned, for sure. For the temperature monitor a board with Wifi or Bluetooth seem like a good fit since you probably don't want to run Ethernet cables to your fridge. But you'll probably need analog input to read the probe... A Raspberry Pi Zero W with an analog to digital add-on would work. There's probably even an add-on specifically for that purpose somewhere out there. And run the server from the Pi itself. Or something like this: www.espruino.com/MDBT42Q (which has analog inputs and Bluetooth) could send the readings to a nearby computer running the server. Bonus points if you run a Tor client on the server and check the status of your fridge over Tor with OrFox! (it would be over-engineering but totally cool and wouldn't require any port-forwarding)

DavyBot v1.0 how would you power the pi inside the fridge?

I was thinking to run the probe into the fridge and set the Pi on top or something. Most of those temperature probes are at the end of a long wire that you could fit between the seal of the door. For instance these ones are cheap and they handle the analog-to-digital): www.google.com/search?q=DS18b20 I'm not sure what kinds of options there are for boards operating at temperatures like that if you wanted to put them on the inside. That would be a fun experiment though! I wonder how long a Raspberry Pi would survive in my freezer...?

DavyBot v1.0 one thing is for sure: no additional cooling needed