An ESP8266 with temp sensor and a script to pull public weather station data with your local temperature readings into a text file is likely the cheapest way.
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I don't think there is any that can record and export data at that price point. I had similar issues in my home and ended up using an Ambient Weather WS-10 display You can get them with various numbers of remote sensors and you a buy additional sensors separately.
I put one sensor outside and scattered the rest throughout my house. That meant I could look at one display and see how the heat or cold was being distributed. It made balancing things a lot easier.
Best of luck!
I agree. I have the WS-2000 that checks all these boxes but it was $299.
What you’re looking for is a 2 channel temperature data logger, but it might be cheaper to just buy 2 single channel ones (a really brief search shows some for like $30)
Take a look at the thermometer/hydrometer from SwitchBot (also available on Amazon). $14 each, but available in packs of 2/4 at a discount.
Nearly all SB products seem to have data export from the app. I have three temp sensors (inside/outside/garage), and also use the smart plug mini to monitor and store power consumption data for devices.
Technology connections recently did a video about refrigerators and used temp logging device which I think you may be looking for. Check out the time 18:20 on this video
Just get 2 of these:
https://www.amazon.com/dp/B09DPCW47P/
Can pair with your phone over Bluetooth, or if you're feeling adventurous, you can spend the rest of your budget on a Raspberry Pi and install Home Assistant on it - it'll pick these things up automatically, and you can start on your quest of understanding your home with one of the best tools out there.
Snag a 4-pack of AAA Eneloops & a charger, and you won't have to worry about disposing of batteries for all of your new sensors.
You can also swap one of their outdoor ones with an indoor one with a screen so you can view at a glance - just look for one that takes AAA batteries.
This is a little convoluted but stay with me:
Weather Underground is (IMHO) the best weather site. A large source of their data is personal weather stations. There should be good advice on their site about these and if not then their users should have done research on them. So I would recommend checking out their site as if you want to set up your own personal weather station.
Probably a bit overkill for your use-case, but I use a bunch of these to monitor and log server rack temp and humidity at various locations at work. External probes optional.