If you randomly selected from the set of all sentient beings throughout time and space, the odds are vanishingly low that you would get the Little Prince as well.
Suppose that he ponders his situation, and concludes that if there were places in the universe where many, many humans can coexist, then it would be unlikely that he would find himself living alone on an asteroid.
If we accept for the sake of an argument that he exists, then someone must be the Little Prince, and be doomed to make incorrect inferences about the representativeness of their situation.
It makes no difference to the Little Prince's observations whether he is the only being in the universe, or in a heavily populated universe where he simply happens to find himself completely isolated.
Similarly, it makes no difference to our observations whether our future contains a mass extinction event or a population explosion. A universe where our future contained a population explosion would contain a vantage point equivalent to our own just as one where our near future contained an extinction event would.
For any individual, the answer to the question "is my situation typical?" is more likely to be yes than no, at least for sufficiently broad definitions of "typical." But that doesn't mean that the answer can't be "no," and unless you define "typical" so broadly as to be meaningless, sometimes it has to be. If you see a possible future event that would render all present and past existences atypical, you can't use anthropic reasoning to determine whether it's likely to happen, because the universes in which the event doesn't happen and the ones where it does still contain the same vantage points prior to it.
The logical conclusion of that version of the anthropic principle is that the universe contains infinitely many copies of us.
...at least not if you accept a certain line of anthropic argument.
Thomas Nagel famously challenged the philosophical world to come to terms with qualia in his essay "What is it Like to Be a Bat?". Bats, with sensory systems so completely different from those of humans, must have exotic bat qualia that we could never imagine. Even if we deduce all the physical principles behind echolocation, even if we could specify the movement of every atom in a bat's senses and nervous system that represents its knowledge of where an echolocated insect is, we still have no idea what it's like to feel a subjective echolocation quale.
Anthropic reasoning is the idea that you can reason conditioning on your own existence. For example, the Doomsday Argument says that you would be more likely to exist in the present day if the overall number of future humans was medium-sized instead of humongous, therefore since you exist in the present day, there must be only a medium-sized number of future humans, and the apocalypse must be nigh, for values of nigh equal to "within a few hundred years or so".
The Buddhists have a parable to motivate young seekers after enlightenment. They say - there are zillions upon zillions of insects, trillions upon trillions of lesser animals, and only a relative handful of human beings. For a reincarnating soul to be born as a human being, then, is a rare and precious gift, and an opportunity that should be seized with great enthusiasm, as it will be endless eons before it comes around again.
Whatever one thinks of reincarnation, the parable raises an interesting point. Considering the vast number of non-human animals compared to humans, the probability of being a human is vanishingly low. Therefore, chances are that if I could be an animal, I would be. This makes a strong anthropic argument that it is impossible for me to be an animal.
The phrase "for me to be an animal" may sound nonsensical, but "why am I me, rather than an animal?" is not obviously sillier than "why am I me, rather than a person from the far future?". If the doomsday argument is sufficient to prove that some catastrophe is preventing me from being one of a trillion spacefaring citizens of the colonized galaxy, this argument hints that something is preventing me from being one of a trillion bats or birds or insects.
And this could be that animals lack subjective experience. This would explain quite nicely why I'm not an animal: because you can't be an animal, any more than you can be a toaster. So Thomas Nagel can stop worrying about what it's like to be a bat, and the rest of us can eat veal and foie gras guilt-free.
But before we break out the dolphin sausages - this is a pretty weird conclusion. It suggests there's a qualitative and discontinuous difference between the nervous system of other beings and our own, not just in what capacities they have but in the way they cause experience. It should make dualists a little bit happier and materialists a little bit more confused (though it's far from knockout proof of either).
The most significant objection I can think of is that it is significant not that we are beings with experiences, but that we know we are beings with experiences and can self-identify as conscious - a distinction that applies only to humans and maybe to some species like apes and dolphins who are rare enough not to throw off the numbers. But why can't we use the reference class of conscious beings if we want to? One might as well consider it significant only that we are beings who make anthropic arguments, and imagine there will be no Doomsday but that anthropic reasoning will fall out of favor in a few decades.
But I still don't fully accept this argument, and I'd be pretty happy if someone could find a more substantial flaw in it.