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.
> 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.
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.