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Imagine that an evil billionaire decided to make Ethereum hard to use by making the gas very high. I'm wondering how expensive this would be to do for various timescales.

For example, manipulating the average gas price over a single minute might be pretty easy. How much would it cost to bring the average gas price up 100%?

Manipulating the average gas price by the same amount over a whole day would probably cost more. How much more?

Probably the answer depends on how many other people are submitting transactions as well. How to characterize this dependency?

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Well it's the free market/miners who decides the average gas price. They basically decide at what gas price they are willing to process a transaction on the network. They can set a threshold for a range of gas prices that they are more likely to process and ignore the rest. Also, smart contracts are flexible in the ways which they allow for gas price modification, some include a gas price limit, others don't, this also affects the average gas price.

There's also a built-in Gas Price Oracle (GPO) in the Ethereum Protocol, this defines the most optimal gas price based on previous block gas usage and recommend future gas prices on the network.

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