Per the wiki, your king's energy gains are reduced while he's hungry - drastically reduced. Between 61-100% hunger, they will regenerate 1% energy every 15 minutes. That is a 90% decrease at a hunger level that's essentially a bit below half.

Well, I think the penalty is a bit steep. Some players only roll over to accept stud requests or breed lions, not even touching explore for certain months out of the year. Perhaps they don't even have any lions to feed, so hunting for food to feed their king isn't something they are doing. I've found plenty of semi-active studs whose hunger bars are in the red zone, and their energy is at a mere 13%. Enough for a single stud attempt before needing to feed their king in order to wait for another regen tick.

To me it doesn't make sense to have it go from 10% energy per 15 minutes straight down to 1% after3 days of not eating. 3 rollovers is technically equivalent to 3 months, but even so, your king is only half starving in the actual game. Yes, it is simple to keep one's king fed. Single carcasses are cheap and whatnot, exploring is free, yada yada. I'm not saying there shouldn't be an energy penalty for having a hungry king, but the sheer drop-off is too much. Especially when the item description of Roasted Lambs and Red Bulls is worded in a confusing manner for some players:

I've seen a fair number of Game Help topics from players wondering why they can't get their Roasted Lamb to work when their king is at 61% hunger, and it has to be even more frustrating to spend 1/8 of your energy item on 4% total energy due to misunderstanding.

My proposal is that, at 61% hunger (when the bar turns red) energy should regenerate at half the normal rate - 5% per 15 minutes. Your king is roughly half hungry, he should get half energy. At 80% hunger, it can be reduced further to 3% energy per 15 minutes. And finally, at 100% hunger, they will only regenerate 1%.

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My other suggestions:
Masterpost here