In CDS settlement, the final price set in the second stage of the auction is used as the recovery rate. CDS pricing usually treats the recovery rate as a constant, unrelated to interest rates or the hazard rate. The higher the recovery rate, the smaller the payment on default, and the lower the CDS spread.
Consider a hypothetical example of a CDS with a notional amount of USD 10 million and a recovery rate of 40%. On default, the protection seller pays the buyer USD 6 million on default.