Response:
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Good question. He seems to have some knowledge of Who God is because he calls Him Lord in Gen 20:4. He seems to believe that God is just because he pleads his innocence and expects God to respond justly. He also does exactly what God tell him to do.

However, the Philistines were polytheists. Therefore, it is possible that Abimelech had no problem believing in Jehovah, but he also had no problem believing in Baal or Dagon. During the time of Eli and Samuel (which, admittedly, is 800 years after Abraham) the Philistines took the Ark of the Covenant after defeating Israel Notice what they said in I Sam 4:6-8

"And when the Philistines heard the noise of the shout, they said, What meaneth the noise of this great shout in the camp of the Hebrews? And they understood that the ark of the LORD was come into the camp. And the Philistines were afraid, for they said, God is come into the camp. And they said, Woe unto us! for there hath not been such a thing heretofore. Woe unto us! who shall deliver us out of the hand of these mighty Gods? these are the Gods that smote the Egyptians with all the plagues in the wilderness"

So, they worshipped their own gods, and believed that Jehovah was God, and believed that there were several gods that helped the Jews. Maybe Abimelech felt the same way.