Effective theories of a scalar $\phi$ invariant under the internal \textitgalileon symmetry $\phi\to\phi+b_\mu x^\mu$ have been extensively studied due to their special theoretical and phenomenological properties. In this paper, we introduce the notion of \textitweakly broken galileon invariance, which characterizes the unique class of couplings of such theories to gravity that maximally retain their defining symmetry. The curved-space remnant of the galileon's quantum properties allows to construct (quasi) de Sitter backgrounds largely insensitive to loop corrections. We exploit this fact to build novel cosmological models with interesting phenomenology, relevant for both inflation and late-time acceleration of the universe.