I doubt it. Recording audio is pretty cheap. Lots of smartphones can now do continuous “wake word” listening just like smart speakers while on battery power.

"Pretty cheap" is still likely an order of magnitude or more than actual standby, and using a small ring buffer to identify a pre-established wake word is similarly going to be cheap compared to actually streaming audio to disk.

If you actually care, this would be pretty trivial to verify using an oscilloscope to observe the energy draw from the battery while different things are said around your device.

I’ve verified it accidentally by using an intelligent alarm with my phone unplugged. The battery drain for eight hours (which includes analysis that would be unnecessary for just recording) was a few percent.