  A recent proposal for gauge-invariant observables in inflation [R. Brunetti
et al., JHEP 1608 (2016) 032] is examined. We give a generalisation of their
construction to general background spacetimes. In flat space, we calculate
one-loop graviton corrections to a scalar two-point function in a general gauge
for the graviton. We explicitely show how the gauge-dependent terms cancel
between the usual self-energy contributions and the additional corrections
inherent in these observables. The one-loop corrections have the expected
functional form, contrary to another recently studied proposal for
gauge-invariant observables [M. B. Fr\"ob, Class. Quant. Grav. 35 (2018)
035005] where this is not the case. Furthermore, we determine the one-loop
graviton corrections to the four-point coupling of the gauge-invariant scalar
field, and the corresponding running of the coupling constant induced by
graviton loops. Interestingly, the $\beta$ function is negative for all values
of the non-minimal coupling of the scalar field to curvature.
