Optical measurements of a nanoscale silicon optomechanical crystal cavity with a mechanical resonance frequency of 3.6 GHz are performed at subkelvin temperatures. We infer optical-absorption-induced heating and damping of the mechanical resonator from measurements of phonon occupancy and motional sideband asymmetry. At the lowest probe power and lowest fridge temperature (T f = 10 mK), the localized mechanical resonance is found to couple at a rate of γ i /2π = 400 Hz (Q m = 9×10 6 ) to a thermal bath of temperature T b ≈ 270 mK. These measurements indicate that silicon optomechanical crystals cooled to millikelvin temperatures should be suitable for a variety of experiments involving coherent coupling between photons and phonons at the single quanta level. The coupling of a mechanical object's motion to the electromagnetic field of a high finesse cavity forms the basis of various precision measurements [1], from large-scale gravitational wave detection [2] to microscale accelerometers [3]. Recent work utilizing both optical and microwave cavities coupled to mesoscopic mechanical resonators has shown the capability to prepare and detect such resonators close to their quantum ground state of motion using radiation pressure backaction [4][5][6][7]. Optomechanical crystals (OMCs), in which band gaps for both optical and mechanical waves can be introduced through patterning of a material, provide a means for strongly interacting nanomechanical resonators with near-infrared light [8]. Beyond the usual paradigm of cavity optomechanics involving isolated single mechanical elements [9,10], OMCs can be fashioned into planar circuits for photons and phonons, and arrays of optomechanical elements can be interconnected via optical and acoustic waveguides [11]. Such coupled OMC arrays have been proposed as a way to realize quantum optomechanical memories [12], nanomechanical circuits for continuous variable quantum information processing [13] and phononic quantum networks [14], and as a platform for engineering and studying quantum many-body physics of optomechanical metamaterials [15][16][17].The realization of optomechanical systems in the quantum regime is predicated upon the ability to limit thermal noise in the mechanics while simultaneously introducing large coherent coupling between optical and mechanical degrees of freedom. In this regard, laser back-action cooling has recently been employed in simple OMC cavity systems [6,18] consisting of a one-dimensional (1D) nanobeam resonator surrounded by a two-dimensional (2D) phononic band gap.