Best to be upbeat here: Badiou’s take on the ‘rebirth of history’ is a topical and elegant response to Fukuyama (he is cited in Chap 2 of LFM), but the idea assumes the ‘end of history’ is coherent, and if not, both ideas fail.
The ‘end of history’ is an idea that LFM takes from incoherence to a series of definitions, thence motivating ‘equal time’ for communism and the more coherent solution to the ‘end of history’ question mark. Still, the idea of the ‘rebirth of history’ has a kind of topical aptness and refers more to the kind of jargon of neocons like Kaplan who mangled the already mangled ‘end of history’ meme, which wasn’t in Hegel.
One innovation in LFM is to consider the ‘end of history’ in light of a ‘finite transition’ model, in which case the ‘end of history’ is a teleological reference to something outside of space-time. Then the incoherence resolves to a pun on the term ‘end’: it is not the normal ‘end’ of a sequence, history, but the ‘end’ in the sense of a goal or directive.