According to Buddhism, how would the end of times look like?
Would it just be the end of a pre-defined period, or would it be something different like the Rapture in Christianity?
This isn't an easy question to answer.
Wikipedia, at least, claims that there is such a thing as Buddhist eschatology, i.e. it says,
There are two major points of Buddhist eschatology: the appearance of Maitreya and the Sermon of the Seven Suns.
I think (I may be wrong) that these two are of two different schools of Buddhism ("Maitreya" being Mahayana, and the "Sermon of the Seven Suns" in the Pali Canon being presumably Theravada).
Here is a translation of The Seven Suns (but there isn't a translation of it on accesstoinsight). I don't know how people interpret it (e.g. whether literally, or as a parable about impermanence, etc).
Ven. Bhikkhu Bodhi (a Theravada monk) writes what might be a mainstream Theravada view, i.e.,
I think that's supported by Suttas such as Gaddula Sutta which says,
At Savatthi. There the Blessed One said: "Monks, from an inconstruable beginning comes transmigration. A beginning point is not evident, although beings hindered by ignorance and fettered by craving are transmigrating & wandering on.
"There comes a time when the great ocean evaporates, dries up, & does not exist. But for beings — as long as they are hindered by ignorance, fettered by craving, transmigrating & wandering on — I don't say that there is an end of suffering & stress.
"There comes a time when Sineru, king of mountains, is consumed with flame, is destroyed, & does not exist. But for beings — as long as they are hindered by ignorance, fettered by craving, transmigrating & wandering on — I don't say that there is an end of suffering & stress.
"There comes a time when the great earth is consumed with flame, is destroyed, & does not exist. But for beings — as long as they are hindered by ignorance, fettered by craving, transmigrating & wandering on — I don't say that there is an end of suffering & stress.
"Just as a dog, tied by a leash to a post or stake, etc.
I think that's saying we can't afford to just assume that suffering will inevitably come to an end some day (a view which might, contrary to Buddhism, assume some form of predetermination or nihilism).
Among the famously unanswered questions is one of which asks whether the world is eternal (see also the Simsapa Sutta referenced in Buddho's answer). I think that these being "unanswered" implies that the Buddha (who was supernaturally omniscient) decided not to answer these questions because the answer wouldn't be helpful (e.g. the answer isn't relevant to the Buddhist goal of ending suffering).
Slightly off-topic but still on the subject of time and of not-time, there is a concept of timelessness, which may or may not interest you. Some of the "formless" meditative states may I guess be timeless too (if time is a property of form); and nirvana is certainly sometimes described as "deathless".
Buddhism isn't Christianity, so it is concerned with a different set of question and has different assumptions.
As for the area of metaphysical speculation, the early Buddhist either explicitly put those questions outside of Buddhism, or assumed that the current assumptions of their day was correct, name that time has been going on for ever, will continue to do so, but go through cycles of things getting better and worse, falling apart and restarting. This is essentially the same world view of anyone in India at that time.
Something unique to Buddhism though, was speculation on how long Buddhism itself would last. All things change and disintegrate including Buddhism, as a movement and coherent set of teachings. Some early predictions thought 500 years, but as that time period came and passed, it was bumped up to a higher number.
The closest thing in the Buddhist system to end times is Mappo, the time of the decline of the Dharma. Many systems of Buddhism believed we were in the time of decline of the Dharma, and therefore we needed new creative ways to practice Buddhism that differed from early Buddhism.
To get a good understanding, read Jan Natiers paper on the topic: https://www.academia.edu/7175816/Buddhist_Eschatology
The "end of times" is not a Buddhist concept so the meaning of the question is ambiguous. If by the end of times you mean the end of the earth, the teaching of the Pali Canon is expressed in the Sermon of the Seven Suns.
This sutta appears to describe a supernova of our sun millions or billions of years in the future, including progressive global desertification, death of vegetation and species, and finally the earth itself being consumed in an expanded solar conflagration that takes tens or hundreds of thousands of years.
If by "end of times" you mean the end of samsara, religious Buddhism refers to this as the goal of the bodhisattva but I am not aware of any canonical references to the end of samsara. Logically, it seems that samsara, having no origin, cannot have an end either.
As long as there is becoming due to ignorance there will be existence in some form. Perhaps not as humans or planet earth, but further becoming is only extinguished through nirvana.
Such questions don't hold much value in Buddhism for they don't lead to liberation.
Once the Blessed One was staying at Kosambi in the simsapa forest. Then, picking up a few simsapa leaves with his hand, he asked the monks, "What do you think, monks: Which are more numerous, the few simsapa leaves in my hand or those overhead in the simsapa forest?"
"The leaves in the hand of the Blessed One are few in number, lord. Those overhead in the simsapa forest are more numerous."
"In the same way, monks, those things that I have known with direct knowledge but have not taught are far more numerous [than what I have taught]. And why haven't I taught them? Because they are not connected with the goal, do not relate to the rudiments of the holy life, and do not lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. That is why I have not taught them.
"And what have I taught? 'This is stress... This is the origination of stress... This is the cessation of stress... This is the path of practice leading to the cessation of stress': This is what I have taught. And why have I taught these things? Because they are connected with the goal, relate to the rudiments of the holy life, and lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. This is why I have taught them.
"Therefore your duty is the contemplation, 'This is stress... This is the origination of stress... This is the cessation of stress.' Your duty is the contemplation, 'This is the path of practice leading to the cessation of stress.'"