Does auxiliary air valve work on a cold start on a hot day?

Basically, my car is a very troublesome little tyke (the intermittent misfire has all but disappeared asides a short episode jumping my not so daily Saturday)

Anyway, so I'm wondering how much an AAV does from cold on a hot day.. cold start in the morning say 15-16 degrees I can hear the AAV slurping away, squeeze the hose and engine bogs down as it should, the only issue is when the car is revved from cold the revs drop lower than idle speed when let off accelerator then pick back up again, once warm revs are pretty stable around 950-1k

However, leaving work in the afternoon at around 25-26 degrees the AAV does nothing, and the car idles a lot lower speed than in the morning along with the revs dropping below idle speed when letting off the accelerator almost dropping to the point the car nearly cuts out, runs fine when warmed up.

Last summer it would sit and not idle and just cut out until warm enough but a test and clean of AAV and it's been much better but still not right

So I know the AAV unit works, opens in the freezer, closes with 12v on it, my theory is that in the morning it's cool enough for it to open itself and close with heatsoak from the engine and that it stays closed all day as the under bonnet temp in the sun is fairly high. Basically I think the wiring to it is bad so it is working mechanically but electrically not so much. Then again maybe it's working fine and the problem lies elsewhere?

Would the 12v power to it make it open with ignition on in the heat?

Is it more likely a WUR or vacuum problem?

Anyone else have any experience of this kind of issue?

Have you checked your idling speed with the ISV (idle stabilising valve) disconnected? When I got my car the revs also dropped and sometimes died when getting off the accelerator. Turned out idle speed was set too low,  so ISV had to constantly try to catch/ correct it when off throttle. When it is warm, it might be idling just high enough not to trigger the ISV. Idle should be set with the ISV disconnected.

Sorry, I have a late cabriolet so it has ISV but same might apply to AAV.

lancia7550 I've removed electrical connection both in the morning and afternoon and it makes no difference I'd guess particularly in the morning because the AAV would already be open. I'm not 100% sure exactly what the electrical element does, I guess closes the valve faster rather than relying on heatsoak from engine? I also don't know what sort of temperatures they would start closing and then be fully closed at?