To Spend Time On
transitive verb, godan verb
過ぎる is to surpass... 過ごす is to spend time on... for some reason. I suppose if you surpass something or someone, you must spend time to do it.
すごす
The reading is the same as 過ぎる. Time to surpass this kanji? Or, maybe it's time to spend time on it instead.
電気料金の支払いが三ヶ月遅れて電気を止められちゃって、あのクソ暑い日をエアコン無しで過ごさなきゃならなかったんだ。
Our electricity stopped because we fell three months behind on our utility bill, so we had to spend the hot day without an air‐conditioner.