The Gould Belt Legacy Survey will map star-forming regions within 500 pc, using HARP (Heterodyne Array Receiver Programme), SCUBA-2 (Submillimetre Common-User Bolometer Array 2) and POL-2 (Polarimeter 2) on the James Clerk Maxwell Telescope (JCMT). This paper describes HARP observations of the J = 3-2 transitions of 13CO and C18O towards Orion A. The 1500-resolution observations cover 5 pc of the Orion filament, including OMC1 (inc. BN-KL and Orion Bar), OMC 2/3 and OMC 4, and allow a comparative study of the molecular gas properties throughout the star-forming cloud. The filament shows a velocity gradient of ~1 km/s /pc between OMC 1, 2 and 3, and high velocity emission is detected in both isotopologues. The Orion Nebula and Bar have the largest masses and line widths, and dominate the mass and energetics of the high velocity material. Compact, spatially resolved emission from CH3CN, 13CH3OH, SO, HCOOCH3, C2H5OH, CH3CHO and CH3OCHO is detected towards the Orion Hot Core. The cloud is warm, with a median excitation temperature of ~24 K; the Orion Bar has the highest excitation temperature gas, at >80 K. The C18O excitation temperature correlates well with the dust temperature (to within 40%). The C18O emission is optically thin, and the 13CO emission is marginally optically thick; despite its high mass, OMC 1 shows the lowest opacities. A virial analysis indicates that Orion A is too massive for thermal or turbulent support, but is consistent with a model of a filamentary cloud that is threaded by helical magnetic fields. The variation of physical conditions across the cloud is reflected in the physical characteristics of the dust cores….continued