  Theoretical descriptions of observable quantities in cosmological
perturbation theory should be independent of coordinate systems. This statement
is often referred to as gauge-invariance of observable quantities, and the
sanity of their theoretical description is verified by checking its
gauge-invariance. We argue that cosmological observables are invariant scalars
under diffeomorphisms and as a consequence their theoretical description is
gauge-invariant, only at linear order in perturbations. Beyond linear order,
they are usually not gauge-invariant, and we provide the general law for the
gauge-transformation that the perturbation part of an observable does obey. We
apply this finding to derive the second-order expression for the observational
light-cone average in cosmology and demonstrate that our expression is indeed
invariant under diffeomorphisms.
