In a manner that is well-pleasing, acceptably — describing how believers should offer worship and live before God. The standard is not human satisfaction but divine pleasure.
Euarestōs in Hebrews 12:28 calls believers to 'offer to God acceptable worship, with reverence and awe.' This word reorients all of life toward divine audience. It doesn't ask 'Does this satisfy me?' but 'Does this please God?' Combined with 'our God is a consuming fire' (Heb 12:29), it teaches that worship is both intimate and awesome — we approach a Father who is also a holy fire.