Somebody: The thing is, a concern for biodiversity is itself largely anthropocentric - which in my opinions does not weaken it one iota. But, really, the birds and spinach don't care for the survival of their species per se. Even from an evolutionary perspective, they "care" in the loosest sense of the term, or behave as if they care, for themselves and their genes.
At any rate, with a rough median projected 3 degree Celsius increase by 2100 by the Intergovernmental Panel on Climate Change, horses and tomatoes will be fine, because we'll keep them around regardless. When scientists talk about species going extinct, to be blunt they're mostly referring to species of insects and plants most of us haven't even heard of and many of whom haven't even been identified, let alone studied. After all, almost all the species present today also were around 12,000 years ago at the end of the last glaciation. The more dramatic fact will be changes in the distribution of species.
I agree with Carrol that the real issue is how the change in climate impacts human civilization. My own opinion is that a much more technologically advanced civilization (capitalist or otherwise) in the late 21st and 22nd centuries will have to devote significant resources into coping with climate change, but that the very fact society will be so much more productive will mean that mortality from average temperature increases will be slight. To wit, heat waves kill in industrialized countries largely to the extent that people don't have air conditioning. If they do, and I wager by the end of the century almost everyone will in warm parts of the world, then few will die from extreme heat.