We have one on central Arizona that never needs to shut down due to heat and definitely does not rely on sea water for cooling. So no, they don’t need to shut down when it gets too hot if they are designed for the heat.

It seems like that's only a question of when it needs to shut down rather than if:

https://www.wired.com/story/nuclear-power-plants-struggling-...

> In the US, the sole desert-based nuclear facility, the Palo Verde plant in Arizona, relies on municipal wastewater rather than rivers or seas, though the facility has struggled with rising costs as more industries compete for limited supplies.

It’s highly ironic that fossil fuels might be creating a problem that cripples nuclear fission going forward. That being said, I’m hoping we manage to scale up direct energy transfer and bypass the need for turbines.

The real irony is that environmental activism using FUD against nuclear energy tech in the ‘70s through early ‘90s crippled it even more.

Because French were cheapening out on cooling towers and water returning back to river would be too hot. It is not problem of fission in general. It is just construction defect of those types of plants.