I don't see a problem in the sleeping mode, as long as the device sends regular updates.
But indeed, a +0,5°C and -0,5°C is way to much.
At least give us a possibility to go to 0,1°C when connected to power, and 0,2°C when in battery mode.
Make room for a disclaimer so the user understands this has a huge impact on battery life.
I love the design of the device, really, but this threshold is making the device useless (for me).