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Transactions would run fine.
It depends. Lightning Network transactions in opened channels? Yes. On-chain transactions? Kind of, because things will take around 10 times longer, and then, after 2016 blocks (so after around 20 weeks, if that would be permanent), the difficulty will adjust to be 4 times easier, then again 2.5 times easier, and since then, the network will function normally.Quote
So more energy use = more security?
Of course. Just download Bitcoin Core and run regtest, even offline. You will have minimal difficulty, minimal energy usage, and all Bitcoin functionality. You will mine all coins in seconds on a CPU. But it will be very insecure, and possible to overwrite instantly, if you will deploy that in such state to the real nodes. 