Proof of Stake systems have no equivalent. They operate by trusting the staking parties are not colluding to create an alternate history. But if they do decide to collude, there is little thermodynamic cost to rewriting history.
It's a fundamentally weaker system, and one that depends on trust. The whole value of proof of work is that it allows you to escape trust.
In a proof of work system, even if you collude you still have to spend a non-trivial sum of electricity to create that alternate history. Electricity that would be making you money via the block reward if you were using it honestly.
It is provably more expensive to reconstruct history in a PoW setting, even when everyone is happily colluding.
In such an adversarial environment proof of stake even has advantages over proof of work. If in a proof of work system a majority of the network starts to double spend or censor transactions, what can you do? You are essentially powerless because any fork (that tries to fix their malicious blocks) can be quickly taken over by the colluding miners. However in a proof of stake system you could simply fork once and destroy/invalidate all the tokens of the malicious miners.
Rewriting a year of history in a proof of stake system takes a couple days on consumer hardware. Rewriting a year of history in Bitcoin is going to require burning a literal hundred million dollars of electricity, if not an order of magnitude more.
In a proof of stake system, typically less than 30% of participants by coin amount actually do the proof of stake. And that's for systems that don't require you to bond coins for months. Exchanges frequently control that many coins. And if an exchange abuses their power, are you going to burn the coins of every single person using the exchange?
It gets worse than that though. You don't need current coins to attack the system. You only need old keys of coins that you sold. After you sell your coins, why wouldn't you sell your keys to an attacker too? If the attacker is paying for the keys, and you sold the coins anyway, there's nothing to lose for you.
Or an exchange can rotate their coins to new addresses and use the old coins to attack the system.
Pow is really an incentive system to keep selfish miners flocking to one chain, by making betting on one chain more profitable than betting on lots of chains at the same time.
The PoS challenge is to do the same without the egregious amounts of wasted energy, by making miners bet their money directly, instead of indirectly through their computers and electricity.
There is absolutely a thermodynamic cost to reversing a transaction in bitcoin. Having the miners run an alternate client doesn't change that.
