Yes, your Cooksy will remain calibrated and paired with your phone when disconnected from power. It will also keep content recording saved locally when using Cooksy Pro.
When restarting, Cooksy will reconnect to the WiFi it was originally set up on. Your phone’s app will need to pair with the Cooksy as well. From there, you’ll just need to have the Cooksy cameras pointed at the cooktop burner you intend to use.