Motivated by the experiment, high mobility of gold atoms on two-dimensional carbon sheets, we examine the ground-state structures, mobility, and clustering formations of small gold clusters (Au_n, n = 1-4) on monolayer alpha-graphyne using first-principles DFT calculations and finite temperature MD simulations. We reveal that Au_n cluster prefers to locate at the center of a hexagon in alpha-graphyne. The binding energy of Au_n on alpha-graphyne increases with increasing the number (n) of gold atoms. Moreover, we predict the step-wise formation of Au_2 out of two pre-adsorbed Au_1 ad-atoms. Likewise, the formation of Au_3 and Au_4 is also considered in the same way. The diffusion energy barrier of Au_1 on alpha-graphyne is found to be only 0.26 eV, indicating the high mobility of gold atoms on alpha-graphyne. Remarkably, the energy required for the cluster formation of gold atoms on alpha-graphyne is about less than 0.2 eV. According to our MD simulations at room temperature (RT), the Au_n cluster is subsequently formed on alpha-graphyne. Considering the high mobility of a single gold atom, the strong binding energy of small gold clusters, and the easy clustering of Au_n at RT on alpha-graphyne, we suggest that alpha-graphyne is a suitable substrate for gold cluster formation.