The number of AES rounds increases with the key length. Why increase the number of rounds at all, and how were these round counts chosen?
There are two reasons:
The 10 rounds for AES-128 seem to be about the lower level of what is (approximately) 128-bit-secure, and 10 rounds for a AES-256-like-cipher would have way below 256 bits of security.
Some quotes from The Design of Rijndael (pdf, see Section 3.5 "The Number of Rounds"):
The addition of four rounds is justified by:
Regarding longer key lengths:
Unfortunately no derivation of this magic 1:32 ratio is given.