Rulebook v3 p14 wrote:
At This Location: These are special powers that are in effect while the location is open. Some of these remain in effect when the location is permanently closed; in that case, they also appear on the back of the location card.
Emphasis mine. The Treacherous Cave does not have an "At This Location" power on the back side of the card, so it stops applying. Compare that with the Temple, which does still have an "At This Location" power on the back of the card.