  We propose a new method for measurements of gravitational acceleration using
a quantum optomechanical system. As a proof-of-concept, we investigate the
fundamental sensitivity for a cavity optomechanical system for gravitational
accelerometry with a light-matter interaction of the canonical `trilinear'
radiation pressure form. The phase of the optical output of the cavity encodes
the gravitational acceleration $g$ and is the only component which needs to be
measured to perform the gravimetry. We analytically show that homodyne
detection is the optimal readout in our scheme, based on the cyclical
decoupling of light and matter, and predict a fundamental sensitivity of
$\Delta g = 10^{-15}$ ms$^{-2}$ for currently achievable optomechanical systems
which could, in principle, surpass the best atomic interferometers even for low
optical intensities. Our scheme is strikingly robust to the initial thermal
state of the mechanical oscillator as the accumulated gravitational phase only
depends on relative position separation between components of the entangled
optomechanical state arising during the evolution.
