A planetary physician would look on biodiversity as a symptom, a response to change. He would recognize that what is a rare species in one state becomes a common one in another. So rich biodiversity is not necessarily something highly desirable and to be preserved at all costs. A red, flushed and sweaty skin is our physiological response to overheating, and the biodiversity of a tropical forest like Amazonia may be the Earth's response to the heat of the present interglacial. Neither of these states is worth preserving as a long-term goal, and evolution would change them into something more stable. I suspect that the capacity to become biodiverse has evolved because, in the real world of Gaia, change is always happening and is usually driven from outside by small alterations in the seating arrangements of the solar system and in the output from the sun. When there is a climate change, dormant seeds, rare plants, or seeds drifting on the wind, or on the feet of birds, have a better or worse chance to grow; if better, they flourish and compete with the native species until they become a stable part of the ecosystem. During the period of competition biodiversity is increased, but it declines again as the ecosystem adapts to the new conditions.
The problem with that theory is that climate changes affect the poles the most and the tropics the least, so while the tropics are warmer than during the ice ages it's not nearly as big a difference as the high latitudes. I would think if his theory about the high biodiversity being caused by changes, the nothern area that were covered by glaciers during the ice age would be the most biodiverse, since the change between thousands of feet of ice and a temperate climate certainly dwarfs the changes the tropics went through.
I personally would suspect a more stable climate would encourage more biodiversity, species could adapt to more specialized niches while a changing climate would encourage more generalist species because they're more adaptable. It seems according to the map the most important factors are warmth and rainfall, those don't explain everything but there's a strong correlation with more rain, more warmth = more biodivresity, mountains also making a difference because more niches are created by diverse terrain.