I shall emphasize those places where Augustine specifically indicates temporary suffering for Christian believers after death:
“[T]hough she [the Church] prays even for the wicked so long as they live, she yet does not even in this world pray for the unbelieving and godless who are dead. For some of the dead, indeed, the prayer of the Church or of pious individuals is heard; but it is for THOSE WHO, HAVING BEEN REGENERATED BY CHRIST, did not spend their life so wickedly that they can be judged unworthy of such compassion, nor so well that they can be considered to have no need of it. As also, AFTER THE RESURRECTION, there will be some of the DEAD to whom, after they have endured the PAINS proper to the spirits of the DEAD, mercy shall be accorded, and acquittal from the punishment of the eternal fire. For were there not some whose sins, though not remitted in this life, shall be remitted in that which is to come, it could not be truly said, “They shall not be forgiven, neither in this world, neither in that which is to come’ (Mt 12:32).” City of God, 21:24 (A.D. 426).
“For one who cultivates this field interiorly and gains his bread, albeit with toil, can suffer this toil up to the end of his life, but after this life he need not suffer. The man who perhaps has not cultivated the land and has allowed it to be overrun with brambles has in this life the curse of his land on all his works, and AFTER THIS LIFE he will have either PURGATORIAL FIRE or eternal punishment. Thus NO ONE escapes this sentence [curse of toil and suffering], but we should act so that we feel its punishment only in this life.” Genesis Defended Against the Manicheans, 2, 20, 30
“The prophet Malachi…predicts the last judgment, saying, ‘Behold, He comes, says the Lord Almighty; and who shall abide the day of His entrance? . . . for I am the Lord your God, and I change not.” From these words it more evidently appears that SOME SHALL IN THE LAST JUDGMENT SUFFER SOME KIND OF PURGATORIAL PUNISHMENTS; for what else can be understood by the word, ‘Who shall abide the day of His entrance, or who shall be able to look upon Him? For He enters as a moulder’s fire, and as the herb of fullers: and He shall sit fusing and purifying as if over gold and silver: and He shall purify the sons of Levi, and pour them out like gold and silver?’ Similarly Isaiah says, ‘The Lord shall wash the filthiness of the sons and daughters of Zion, and shall cleanse away the blood from their midst, by the spirit of judgment and by the spirit of burning’ (Isaiah 4:4). Unless perhaps we should say that they are cleansed from filthiness and in a manner clarified, when the wicked are separated from them by penal judgment, so that the elimination and damnation of the one party is the purgation of the others, because they shall henceforth live free from the contamination of such men. But when he says, ‘And he shall purify the sons of Levi, and pour them out like gold and silver, and they shall offer to the Lord sacrifices in righteousness; and the sacrifices of Judah and Jerusalem shall be pleasing to the Lord,; he declares that those who shall be purified shall then please the Lord with sacrifices of righteousness, and consequently they themselves shall be PURIFIED FROM THEIR OWN UNRIGHTEOUSNESS which made them displeasing to God. Now they themselves, when they have been purified, shall be sacrifices of complete and perfect righteousness; for what more acceptable offering can such persons make to God than themselves? But this question of purgatorial punishments we must defer to another time, to give it a more adequate treatment. By the sons of Levi and Judah and Jerusalem we ought to understand THE CHURCH herself, gathered not from the Hebrews only, but from other nations as well; nor such a Church as she now is, when ‘if we say that we have no sin, we deceive ourselves, and the truth is not in us’ (1 John 1:8), but as she shall then be, PURGED BY THE LAST JUDGMENT as a threshing-floor by a winnowing wind, and those of her members who need it being CLEANSED BY FIRE….” City of God, 20:25, “Of Malachi’s Prophecy, in Which He Speaks of the Last Judgment, and of a Cleansing Which Some are to Undergo by Purifying Punishments” (A.D. 426)
“…will any man say this time of faith can be placed on an equal footing with that consummation when they who offer sacrifices in righteousness shall be PURIFIED BY THE FIRE OF THE LAST JUDGMENT? And consequently, since it must be believed that AFTER SUCH A CLEANSING THE RIGHTEOUS SHALL RETAIN NO SIN, assuredly that time, so far as regards its freedom from sin, can be compared to no other period, unless to that during which our first parents lived in paradise in the most innocent happiness before their transgression…. For the animals selected as victims under the old law were required to be immaculate, and free from all blemish whatever, and symbolized holy men free from all sin, the only instance of which character was found in Christ. As, therefore, AFTER THE JUDGMENT THOSE WHO ARE WORTHY OF SUCH PURIFICATION SHALL BE PURIFIED EVEN BY FIRE, and shall be rendered thoroughly sinless, and shall offer themselves to God in righteousness, and be indeed victims immaculate and free from all blemish whatever…. Then, with reference to those who are worthy not of cleansing but of damnation….” City of God, 20:26 (A.D. 426)
“But TEMPORARY PUNISHMENTS are suffered by some in this life only, by others AFTER DEATH, by others both now and then; but all of them before that last and strictest judgment. But of those who suffer TEMPORARY PUNISHMENTS AFTER DEATH, all are not doomed to those everlasting pains which are to follow that judgment; for to some, as we have already said, what is not remitted in this world is remitted in the next, that is, they are not punished with the eternal punishment of the world to come.” City of God, 21:13 (A.D. 426)
“We shall then ascertain who it is who can be saved by fire (1 Cor 3:15), if we first discover what it is to have Christ for a foundation…. Whoever, then, has Christ in his heart, so that no earthly or temporal things— not even those that are legitimate and allowed— are preferred to Him, has Christ as a foundation. But if these things be preferred, then even though a man seem to have faith in Christ, yet Christ is not the foundation to that man; and much more if he, in contempt of wholesome precepts, seek forbidden gratifications, is he clearly convicted of putting Christ not first but last, since he has despised Him as his ruler, and has preferred to fulfill his own wicked lusts, in contempt of Christ’s commands and allowances….. For so long as he does not prefer such an affection or pleasure to Christ, CHRIST IS HIS FOUNDATION, though on it he builds wood, hay, stubble; and therefore he shall be saved as by fire. For the FIRE OF AFFLICTION shall burn such luxurious pleasures and earthly loves, though they be not damnable, because enjoyed in lawful wedlock. And of this fire the fuel is BEREAVEMENT, and all those CALAMITIES which consume these joys. Consequently the superstructure will be LOSS to him who has built it, for he shall not retain it, but shall be AGONIZED BY THE LOSS of those things in the enjoyment of which he found pleasure. But by this fire he shall be saved through virtue of the foundation, because even if a persecutor demanded whether he would retain Christ or these things, he would prefer Christ…. But some of them it [fire at last judgment] so proves that it does not burn and consume the structure which is found to have been built by them on Christ as the foundation; while others of them it proves in another fashion, so as to burn what they have built up, and thus cause them to SUFFER LOSS, WHILE THEY THEMSELVES ARE SAVED because they have retained Christ, who was laid as their sure foundation, and have loved Him above all. But if they are saved, then certainly they shall stand at the right hand, and shall with the rest hear the sentence, ‘Come, you blessed of my Father, inherit the kingdom prepared for you;’ and not at the left hand, where those shall be who shall not be saved, and shall therefore hear the doom, ‘Depart from me, you cursed, into everlasting fire.'” City of God, 21:26, “What It is to Have Christ for a Foundation, and Who They are to Whom Salvation as by Fire is Promised,” (A.D. 426)
“Nor can it be denied that the souls of the dead are benefited by the piety of their living friends, who offer the sacrifice of the Mediator, or give alms in the church on their behalf. But these services are of advantage only to those who during their lives have earned such merit, that services of this kind can help them. For there is a manner of life which is neither so good as not to require these services after death, nor so bad that such services are of no avail after death; there is, on the other hand, a kind of life so good as not to require them; and again, one so bad that when life is over they render no help. Therefore, it is in this life that all the merit or demerit is acquired, which can either RELIEVE OR AGGRAVATE A MAN’S SUFFERINGS AFER THIS LIFE. No one, then, need hope that after he is dead he shall obtain merit with God which he has neglected to secure here. And accordingly it is plain that the services which the church celebrates for the dead are in no way opposed to the apostle’s words: ‘For we must all appear before the judgment-seat of Christ; that every one may receive the things done in his body, according to that he has done, whether it be good or bad;’ for the merit which renders such services as I speak of profitable to a man, is earned while he lives in the body. It is not to every one that these services are profitable. And why are they not profitable to all, except because of the different kinds of lives that men lead in the body? When, then, sacrifices either of the altar or of alms are offered on behalf of all the baptized dead, they are thank-offerings for the very good, they are propitiatory offerings for the not very bad, and in the case of the very bad, even though they do not assist the dead, they are a species of consolation to the living. And where they are profitable, their benefit consists either in obtaining a full remission of sins, or at least in making the condemnation more tolerable.” Enchiridion Ch. 110