2x AA. Evehome asks that you use "good" batteries, whatever that means. We have been using the devices for about a year now and according to the graphical display of the charge level in the Eve app, the batteries should last another year. You should be able to get good batteries for about 2 euros each. But it is always a cost factor.

I use rechargeable batteries and they last me plus/minus one year. The app indicates in good time when it is time to change the batteries.

The information about the batteries is somewhat hidden in the data sheet. There are 2 x AA batteries, which were included in our delivery.
We have been using the thermostats for a good year and so far they are working perfectly.