If the internet service goes down:
If your home loses local internet connectivity or your local internet connection is weak, you will not have the ability to control your smart devices via the mobile app until a sufficient connection is restored or the hub comes back online. However, you can still control your devices manually (typing your code into the lock, adjusting the temperature from the thermostat wall unit and so on) if the internet is disconnected.
If the power goes out:
If the power goes out, you will NOT be locked out of your home. Your smart door lock runs off of batteries and any codes that have already been added to your lock will still work.You will not, however, be able to remotely control the lock (or any other devices in the app) or add and delete access codes until the power and the hub come back online. The hub is how the app communicates with your paired devices, and it is powered through an electrical wall outlet.
Your thermostat and HVAC run off electricity as well, so similar to power outages before smart home devices, you will not be able to use your thermostat and HVAC until your power comes back on. The thermostat does have battery backup, so the screen will still display the current temperature and settings.