The present invention relates to a wireless access system supporting an unlicensed band, various methods for adjusting a contention window size, methods for performing a carrier sensing process in a single engine and a multi-engine, and an apparatus supporting the same. As one embodiment of the invention, a method for adjusting a contention window size (CWS) in a wireless access system supporting an unlicensed band may comprise: a step of setting a contention window size (CWS) necessary for performing a channel access process for each of priority classes; a step of performing a channel access process for a particular priority class in a channel in an unlicensed band on the basis of the set CWS; a step of performing a downlink transmission related to the particular priority class if it is determined that the channel in the unlicensed band is in an idle state as a result of the channel access process; and a step of making adjustments for all CWS' of priority classes if a triggering event, which relates to a downlink transmission and increases or reduces the CWS for the particular priority class, occurs.