We prove a large deviation principle for the expectation of macroscopic
observables in quantum (and classical) Gibbs states. Our proof is based
on Ruelle-Lanford functions [20, 34] and direct subadditivity arguments,
as in the classical case [23, 32], instead of relying on G¨artner-Ellis theorem,
and cluster expansion or transfer operators as done in the quantum case
in [21, 13, 27, 22, 16, 28]. In this approach we recover, expand, and unify
quantum (and classical) large deviation results for lattice Gibbs states. In
the companion paper [29] we discuss the characterization of rate functions
in terms of relative entropies.