Aren’t those who face their second death after being judge separated from God in the end?
Regarding the penalty phase, I believe that as people are cast into the “lake of fire” and burned up, they will find themself in either, or both of these situations:
Weeping: they will suffer tremendous regret that they blew it, before they are quickly destroyed.
Gnashing of teeth: they will be angry at God and rebellious to the bitter end before they are snuffed.