Does the NAD 326 Bee Integrated amp run hot

If you can actually keep your hand on the chassis even though it is really hot. Then it is NOT 'too hot'.
If you cannot keep your hand on it at all or get burns, then yeah, it is really too hot.
Hot is very subjective. If the room is warm, say 80 F then the amp is going to be hotter too.
Some amps which are biased more toward class "A" run much hotter than ones not so heavily biased.

The thermal protection hasn't triggered on mine but it does run quite warm....But sounds very ,very good...

Mine runs warm and the thermal protection has never has been triggered. I keep it well ventilated. It's driving Harbeth P3ESR's quite nicely.

Mine runs hot too. Running small mini monitors.
Brian