Inflation and the second law of thermodynamics

Question

Some people, in particular Roger Penrose, claim that an early cosmic inflation can not be the true explanation for some well known observed features of our universe, such as its flatness, smoothness, and lack of observed monopoles for example.

For some reasons, I have a hard time understanding his (counter) arguments. One of the main objections seems to go along the line that if the early cosmic inflation indeed flattens and smoothens any initially present irregularities of our universe, this would go against the second law of thermodynamics.

How is the second law of thermodynamics maintained during the early inflationary phase of the universe? Did the entropy have to increase during inflation, or was the process some kind of adiabatic and the entropy increased only when reheating took place?