Refinance closing timeline

[Ilya Plugovoy]

I am in the process of a refinance. The bank said we will close next week or the week after that. I have a payment due September 1st. If the refinance closes before that date, I wouldn't have to make the payment but there is a chance it will close first few days of September. If I just make an early payment on my mortgage, am I losing the interest portion if we do close before the end of the month? Or regardless the bank will collect their interest for the month of August and it will be all calculated in the payoff?