For more general alphabets, there is a beautiful geometric approach of Csiszár that builds on his theory of information projections, provided one is interested in the probability that an empirical measure based on an i.i.d. sequence lies in a convex set of probability measures. See his 1984 paper "Sanov property, generalized I-projection, and a conditional limit theorem".

There is indeed a geometric way of understanding large deviation results using information theory. For finite alphabets, the classical "method of types" can be used to prove Sanov's theorem (the prototypical result of large deviation theory); this can be found either in the textbook "Elements of Information Theory" by Cover and Thomas, or in the 1998 survey article "The Method of Types" by Csiszár, or in this blog post by Ramon van Handel: blogs.princeton.edu/sas/2013/10/10/lecture-3-sanovs-theorem