It's a range and species have adapted to live in those ranges from desert to arctic. It is true that higher diversity is in temperate and tropical climates but that does not mean that rising temperatures in those zones will equate to even higher diversity or that increases in hot or colder climates will automatically equate to higher biodiversity. Most species have evolved to exist in the relatively narrow ranges to which they have adapted. Or have over millennia developed adaptations for ranges of temperatures like migration and hibernation. Changes in temperatures will affect these behaviors and affect diversity. We are already seeing this in marginal climates.