This is a bit quirky: For a very long time I've found Stephen Hawking's evaporating small black holes a lot more reasonable and intuitive than large black holes.
The main reason is that gravity is relative only if your gravity vectors are all parallel. When that is true, you can simply accelerate along with the field and have a perfectly relativistic frame going for you.
Not so if your gravity vectors angle in towards each other, as is particularly true for very small black holes. In that case the energy inherent in the space around the hole becomes quite real and quite hot, and that's regardless of whether you have matter in the mix or not. (Hopefully that's fairly intuitive to everyone in this group?)
So, how can the space immediately surrounding a tiny black hole not be incredibly hot? By its very geometry it must be absolutely bursting with energy due to the non-parallel intersection of extremely intense gravity vectors. So, the idea of that energy evidencing itself in the creation of quite real particles outside of the event horizon seem almost like a necessity, a direct consequence of the energetic structure of space itself.
So, that's really the basis for my question: Isn't the curvature of space a better way to understand its entropy adding up the surface area of a black hole?
By focusing on curvature, all space has entropy, not just the peculiar variety of space found on event horizons. Flat space maximizes entropy, while the insanely curved space near a microscopic black hole maximizes it. I also like this because if you get right down to it, entropy is really all about smoothness, in multiple forms.
So: Is the inverse of space curvature considered an entropy metric? If not, why not? What am I missing?