It happened again. In Chapter 26 we see Yitzhak going to the city of G'rar. He followed in his father's footsteps and told the people (and Avimelech the king) that Rivka is his sister so they would not kill him and take her. Avimelech played sleuth and figured out that they are husband/wife and not brother/sister.
Avimelech chastised him for the lie saying somebody could have possibly approached Rivka and we all would have been punished for it. "How could you do this to us?" He then told his people to leave Yitzhak and Rivka alone. They continued to live there and prospered.
This is consistent with how Avimelech reacted with Avraham. When Avraham pulled the same trick of saying Sarah is his wife, Avimelech nearly got killed for it and when he found out he gave Avraham great wealth and a choice of land to live on.
Yet when Avraham pulled the same trick on Pharoh, the Egyptians threw him out of Egypt. They gave him money and told him to get lost.
I find the different reactions interesting and telling regarding their attitudes. At the time Avraham went down to Egypt we are told that Egypt had a decadent society and therefore Avraham had great fear for his life. That might be consistent with the Egyptians throwing Avraham out after having duped them. Yet Avimelech had a positive attitude and understood Avraham and Yitzhak's concerns and allowed them to continue living in his country.