There is much science behind the Golden Ratio (Golden Section), of which the Rule of Thirds is a simple approximation. The ratio (known as phi) of approximately 1.618 occurs time and time again in nature and mathematics.
The question is if there is any science behind the aesthetics of that ratio. Sure it occurs in nature, the Greeks thought the ratio was beautiful, etc., but is there any actual proof behind it? It would seem very difficult to scientifically prove something that is subjective, and research would seem to be inconclusive.
A study of famous paintings concludes:
A statistical study on 565 works of art of different great painters
was done and it was calculated the ratio of the 2 sides of a
paintings. Assuming that all the painters under discussion enter in a
statistics with equal weights it is shown that the average value
obtained for the ratio of the sides is 1.34. This value, determined
experimentally is significantly different from the value of the Golden
Section F=1.618, which is a theoretical ratio, obtained from an
abstract, mathematical theory, which supposedly ought to impress on a
painting a supreme harmony
Fechner, Godkewitsch, and Benjafield conducted studies where subjects were asked to rank various rectangles on attractiveness. These studies have contradicted each other, but overall there seems to be a preference towards rectangles that have sides whose lengths are near the golden ratio.
Dr Mario Livio, a scientist and art fanatic, has written an article on the subject, and concludes:
The history of art has nevertheless shown that artists who have
produced works of truly lasting value are precisely those who have
departed from any formal canon for aesthetics