[AnimalMuppet]

Snarky answer: They sleep on piles of money.

Less snarky: They have different anxieties. They don't worry about losing their house or their job. They don't worry about the monthly bills or an unexpected medical event. Instead, they worry about being back in the place where they have to worry about those things. (I think. I'm not, alas, in a position to know first-hand what billionaires are anxious about.)

So they spread their money around. "Divide your fortune into seven, or even into eight, for many are the mischances of the world" is good advice. (It's from Proverbs, in the Bible.) Billionaires do this, so that they can comfortably survive any particular investment going horribly bad.

Then, I suspect, the more aware billionaires worry about their effect on people. They can throw money at friends' and relatives' problems, but that may actually cause more harm than good. I don't know how they deal with that - maybe by having a fixed policy of what they will and won't do, and not deviating from it no matter how good the story sounds.

[Cakez0r]

I imagine the anxieties of billionaires are more existential. For the average person, a huge chunk of your time and energy is spent on obtaining enough wealth to support your lifestyle. Life is about providing for you and your loved ones, and how you can provide more.

Without that purpose to pursue, a billionaire might feel anxious about what to do with their life.