Purgatory

Purgatory is an intermediate state where the souls of the dead can undergo purification before entering the Kingdom of Paradise. Purgatory is also described as a place where "neutral souls" can be housed temporarily until the Ancient of Days issues judgment. Souls in Purgatory can still be cast out into the Outer Darkness.

Half-Bloods and other pagans in Purgatory
Purgatory offers half-bloods, legacies, former Einherjar, and magicians who have pissed off the pagan gods a chance to escape their wrath and enter heaven.

It's important to note that renouncing kinship and loyalty to the Pagan gods is an nonnegotiable precondition to entering heaven and that the purification process of Purgatory erases pagan magic.

Purgatory is protected by the Angels of Paradise. Although the Angels will grant asylum to repentant pagans they are also sadistic towards these pagan and humiliate them "just to make sure their repentance is sincere".

The souls of pagan heroes in life are indistinguishable from mortals by the time they arrive in the Kingdom of Paradise.

Purgatory is constantly under siege by the Angels of Darkness. At one point of the Angels of Darkness conquered Purgatory so the Angels of Paradise summoned Percy Jackson to help them liberate the area.