Templates cache is not writable

I keep getting this error message in the cache report. When I click “check again,” it says the templates cache is writable, but that doesn’t stick. Every time I go back to the cache report, I get the same error. I’m wondering if this is what caused the time zones on my calendar to be four hours off after the last update. Weirdly, this also seems to be interfering with my theme customizer. The customizer will not load the live preview while the cache report reads “not writable.”

https://www.ads-software.com/plugins/all-in-one-event-calendar/