I came up with a simple hardware block diagram and chose a few parts that will make up the main system components.  I think I should remove low cost from the title because I don't think it will be as inexpensive as I was hoping.

I thought about various configurations that the system could be in and decided that I wanted to put everything in a single package.  This may cause issues in the future as batteries do not run well in cold and I'm not sure how well wifi will penetrate the walls of a fridge.  I will try to put everything in a single package for easy installation and if there are issues then I will try to work around it or change the design.

I've attached a basic hardware block diagram and I would like the system to following way:

• Achieve lowest power possible by turning things off or entering sleep modes
• Wake up periodically to test temperature and battery voltage and store data to memory.  If there is a temperature issue then wake up on a higher frequency period.
• When the door opens, I want the system to wake up and display the current temperature
• Access to data through a phone app
• phone app to change alarm thresholds for temperature and battery discharge level

I chose to use an RTD over a cheaper type of temperature sensor to have some decent accuracy since this is for food safety.

I have decided to use the ATwinc1500 wifi module over ESP8266 and its derivatives due to a lower current consumption in idle mode.

I will probably use the ATmega328pb MCU since I already have libraries made and I am comfortable with it.  For a final product the MCU should probably be optimized for what the system actually needs.