Can I use Eneloop batteries in my thermostat?
Eneloop batteries are NiMH which has a normal voltage of 1.2V vs. 1.5V for an alkaline batteries. Your thermostat uses two batteries so the Eneloops will generate 2.4V instead of 3V for fresh alkalines. At a worst case, the thermostat will not work (or report low batteries with fully charged batteries). Best case, the batteries will last for a short time.

As for specifying brand name alkalines, it's probably a partnership as @ratchetfreak suggested. They may also be trying to avoid customer dissatisfaction when cheap, off-brand batteries work poorly or last for a short time.