We study the 850um emission in X-ray selected AGN in the 2 sq-deg COSMOS field using new data from the SCUBA-2 Cosmology Legacy Survey. We find 19 850um bright X-ray AGN in a high-sensitivity region covering 0.89 sq-deg with flux densities of S850=4-10 mJy. The 19 AGN span the full range in redshift and hard X-ray luminosity covered by the sample - 0.7<z<3.5 and 43.2<log10(LX) <45. We report a highly significant stacked 850um detection of a hard X-ray flux-limited population of 699 z>1 X-ray AGN - S850=0.71+/-0.08mJy. We explore trends in the stacked 850um flux densities with redshift, finding no evolution in the average cold dust emission over the redshift range probed. For Type 1 AGN, there is no significant correlation between the stacked 850um flux and hard X-ray luminosity. However, in Type 2 AGN the stacked submm flux is a factor of 2 higher at high luminosities. When averaging over all X-ray luminosities, no significant differences are found in the stacked submm fluxes of Type 1 and Type 2 AGN as well as AGN separated on the basis of X-ray hardness ratios and optical-to-infrared colours. However, at log10(LX) >44.4, dependences in average submm flux on the optical-to-infrared colours become more pronounced. We argue that these high luminosity AGN represent a transition from a secular to a merger-driven evolutionary phase where the star formation rates and accretion luminosities are more tightly coupled. Stacked AGN 850um fluxes are compared to the stacked fluxes of a mass-matched sample of K-band selected non-AGN galaxies. We find that at 10.5<log10(M*/M0)<11.5, the non-AGN 850um fluxes are 1.5-2x higher than in Type 2 AGN of equivalent mass. We suggest these differences are due to the presence of massive dusty, red starburst galaxies in the K-band selected non-AGN sample, which are not present in optically selected catalogues covering a smaller area.