You can use the unarmed strike damage of a monk 4 levels lower than your character level (minimum 1st) instead of the base damage for the chosen weapon. Ascetic Strike functions in all other ways as the brawler’s close weapon mastery class feature.
When worn, this simple brown robe confers great ability in unarmed combat. If the wearer has levels in monk, her AC and unarmed damage is treated as a monk of five levels higher. If donned by a character with the Stunning Fist feat, the robe lets her make one additional stunning attack per day. If the character is not a monk, she gains the AC and unarmed damage of a 5th-Level monk (although she does not add her Wisdom bonus to her AC). This AC bonus functions just like the monk’s AC bonus.
At level 20 would a character using Ascetic Strike use his level 20 unarmed combat damage on his weapon?
Nope. Ascetic Strike doesn't use effective monk level, it takes your character level -4, looks at the Monk table in the CRB, and uses that value. It doesn't matter if something affects your effective monk level (Monastic Legacy, Monk's Robe, actual levels in Monk), because the feat doesn't use that for any calculation (and doesn't say it stacks with Monk levels).
Of course, if you do have levels in Monk (which I presume from your other thread), you don't need Ascetic Strike, as Ascetic Style already lets you apply your full improved unarmed strike damage to the weapon.