What happens to the Capitol City during a Temporal Anomaly?
The Objective Location Card is unaffected.
The rules differentiate between regular Location cards, and the Objective Location card. (Page 2) Game Contents

35 Card Location Deck
1 Double-Sided Objective Location Card

You clearly cannot turn it over, since the rules tell you don't use the Historic Monument in the same game as the Capitol City (page 7).

(On the back side of this Objective Location card is the Historic Monument which can be used as an alternate Objective in later games once you have gotten a handle of the rules).