Nicene and Post-Nicene Fathers, Vol. III:
Doctrinal Treatises of St. Augustin: Chapter 93

Chapter 93.—Both the First and the Second Deaths are the Consequence of Sin. Punishment is Proportioned to Guilt.

And neither the first death, which takes place when the soul is compelled to leave the body, nor the second death, which takes place when the soul is not permitted to leave the suffering body, would have been inflicted on man had no one sinned. And, of course, the mildest punishment of all will fall upon those who have added no actual sin, to the original sin they brought with them; and as for the rest who have added such actual sins, the punishment of each will be the more tolerable in the next world, according as his iniquity has been less in this world.