A colony that ignores its food is worrying, but it is often normal. Here is what to check, in order.

In autumn, European species slow down before hibernation: they eat much less protein, then almost nothing. This is normal. The hibernation calculator tells you when to start.
Below about 20 °C most colonies eat little and the brood grows slowly. Above 30 °C they may also hide. Aim for 22–28 °C depending on the species.
Protein is mostly used to feed the larvae. With no larvae (only eggs or cocoons), a colony often refuses insects and only takes sugar. Protein demand returns with new larvae.
Workers with a round, swollen abdomen have already filled up. An overfed colony leaves leftovers: reduce the amounts.
A colony that has just been moved, handled or exposed to light may stop foraging for a few days. Leave it quiet and dark.
If the colony stops eating and workers die in numbers, the queen stops laying in the middle of the season, or the brood disappears, look for another cause: tube water gone, heat, toxic product, mold (see mold in the nest).
A drop in appetite for a few days or weeks, with no deaths, is almost always one of these normal causes.
Often because there are no larvae: protein is mostly used to feed them. The colony then only takes sugar.
Yes. European species slow down before hibernation.
Several weeks if it has water, especially when cool. Water is more urgent than food.
Amounts and timings are average guidelines, not rules. Always adjust to what you see in your own colony (leftovers, appetite, brood).