[T]he kingdom of darkness, as it is set forth in these and other places of the Scripture, is nothing else but a confederacy of deceivers that, to obtain dominion over men in this present world, endeavour, by dark and erroneous doctrines, to extinguish in them the light, both of nature and of the gospel; and so to disprepare them for the kingdom of God to come. (Leviathan ch. 44)
While Hobbes's Leviathan gets a lot of attention for its discussion of the state of nature and social contract, the second half of the book gets quite a bit less attention; in those chapters, Hobbes first looks at the Kingdom of God, or Christian Commonwealth, and then discusses the Kingdom of Darkness, which he defines as above. He identifies four ways in which this Kingdom of Darkness is furthered:
(1) "abusing and putting out the light of the Scriptures"
(2) "introducing the demonology of the heathen poets"
(3) "mixing with the Scripture diverse relics of the religion, and much of the vain and erroneous philosophy of the Greeks"
(4) "mingling with both these, false or uncertain traditions, and feigned or uncertain history"
The most serious form of (1) is the claim that the Church is the Kingdom of God. The Kingdom of God, as Hobbes understands it, is a political organization, instituted by Moses, for the sake of governing the Jews, which ended with the election of Saul as king. This Kingdom will, according to the Prophets, one day be restored. In his first coming, Jesus began to lay the groundwork for its actual restoration; and since he hasn't come again, there is no Kingdom of God at all (except in the weak sense that we Christians are, as it were, signing up for the future Kingdom of God through the ritual of baptism). The people who are most egregious in making this identification are, of course, the Catholics. Most of Hobbes's discussion of the Kingdom of Darkness is a discussion of the participation of Catholics in the "confederacy of deceivers." Hobbes sees the Catholic Church as a serious threat to his account of sovereign, wherein the sovereign is chosen by the people to exercise the fullness of power. Popes and Catholic councils do not treat heads of state as Hobbesian sovereigns. Hobbes notes with some distaste the statement of the Fourth Lateran Council, "That if a king, at the pope's admonition, do not purge his kingdom of heresies, and being excommunicate for the same, do not give satisfaction within a year, his subjects are absolved of the bond of their obedience." In other words, under certain conditions, the Church can abolish the power of sovereign. Claims like this darken the minds of people and lead to religious wars, Hobbes thinks, because it makes it impossible for ordinary people to tell who they should obey. The Church has an alternative prince (the pope), it has alternative laws (canon), it has an alternative government (the clergy). His concern goes even so far as to worry that the Church has the means of raising an army:
From the same it is that in every Christian state there are certain men that are exempt, by ecclesiastical liberty, from the tributes and from the tribunals of the civil state; for so are the secular clergy, besides monks and friars, which in many places bear so great a proportion to the common people as, if need were, there might be raised out of them alone an army sufficient for any war the Church militant should employ them in against their own or other princes.
This is a very odd worry; perhaps it is a version of Jesuit fears.
Another abuse of Scripture that contributes to the Kingdom of Darkness is "the turning of consecration into conjuration, or enchantment"; he has Catholic sacramental theology in view. Yet another is the claim that the soul is immortal (Hobbes thinks that we lost eternal life at the Fall and it was only restored by Christ for those who enter into the Kingdom of God; he doesn't, I think, believe in hell). There are quite a few things that go with this:
This window it is that gives entrance to the dark doctrine, first, of eternal torments, and afterwards of purgatory, and consequently of the walking abroad, especially in places consecrated, solitary, or dark, of the ghosts of men deceased; and thereby to the pretences of exorcism and conjuration of phantasms, as also of invocation of men dead; and to the doctrine of indulgences; that is to say, of exemption for a time, or for ever, from the fire of purgatory, wherein these incorporeal substances are pretended by burning to be cleansed and made fit for heaven.
The second way in which the Kingdom of Darkness is furthered is by introducing the demonology of the heathen poets, and one example of this is the claim that there are incorporeal spirits. So Catholic angelology and demonology takes a beating. Another example is "the worship of images," and Catholic iconography is in view. Ditto with the canonizing of saints:
The first that ever was canonized at Rome was Romulus, and that upon the narration of Julius Proculus, that swore before the Senate he spoke with him after his death, and was assured by him he dwelt in heaven, and was there called Quirinus, and would be propitious to the state of their new city: and thereupon the Senate gave public testimony of his sanctity. Julius Caesar, and other emperors after him, had the like testimony; that is, were canonized for saints: for by such testimony is canonization now defined, and is the same with the apotheosis of the heathen.
Indeed, as is clear from the other examples Hobbes notes, just about the whole of Catholic liturgy is implicated.
The third and fourth ways way in which the Kingdom of Darkness is furthered are through vain philosophy and superstitious traditions. Needless to say, scholastic thought (with an honorable mention to the Talmudic discussions of Jewish rabbis) comes under the gun here. He also mentions hagiography.
In other words, although 'Kingdom of Darkness' has a larger extension than 'Catholic Church' (all non-Christians are members of the Kingdom of Darkness), the attack on the Kingdom of Darkness is undeniably an attack on Catholicism.
The Kingdom of Darkness, as Hobbes sees it, expands through a will to power. By spreading around these ideas, which obscure the simplicity of Scripture and Hobbesian civil philosophy, priests and the like gain power over their fellow men.
He ends the discussion in an ominous tone:
It was not therefore a very difficult matter for Henry the Eighth by his exorcism; nor for Queen Elizabeth by hers, to cast them out. But who knows that this spirit of Rome, now gone out, and walking by missions through the dry places of China, Japan, and the Indies, that yield him little fruit, may not return; or rather, an assembly of spirits worse than he enter and inhabit this clean-swept house, and make the end thereof worse than the beginning? For it is not the Roman clergy only that pretends the kingdom of God to be of this world, and thereby to have a power therein, distinct from that of the civil state.
It's interesting that anti-Catholic polemic takes up such a significant place in Hobbes's political philosophy; he spends an immense amount of space attacking the Catholic Church both qua Catholic and qua the perfect summation of priestcraft. It's difficult to say more; one area in which Hobbes scholarship is very weak is precisely in this area. There are a number of interesting things that have been noted -- e.g., that Hobbes clearly has in view on some points the censuring by the University of Paris of Gassendi's philosophy, and that this entire section is supposed to illuminate in some way Hobbes's reflections on the Christian Commonwealth -- but it has been difficult for Hobbes scholars to provide a unified view of the whole. It isn't clear what, precisely, the role of the Kingdom of Darkness critique plays in his political philosophy at large, in part because it's obscure in itself, but in part because there hasn't been enough work done on the matter.