Paul uses the verb “eagerly await” a number of times.  What is he waiting for?  He awaits the Savior from heaven (Philippians 3:20), the apocalypse of the Lord Jesus (1 Corinthians 1:7), the revelation of the sons of God (Romans 8:19), the adoption of sons and redemption of the body (8:23).  The author of Hebrews says that we eagerly await Jesus to appear a second time (9:28).  When Jesus comes, at the apocalypse of Jesus, secret sins will be revealed, and the hidden righteous-judgment (dikaiokrisia) will be revealed (Romans 2:5).

But then in Galatians 5:5 Paul says that we eagerly await the hope of righteousness by faith (dikaiosune ek pisteos).

This is the hope of Christians: That Jesus the Savior would come, that our adoption would be revealed in the redemption of the body, that Jesus would unveil Himself.  It is also a hope for justification by faith, which is to say a hope for God’s righteous judgment by the Faithful One, Jesus.