A first-principles-based effective Hamiltonian technique is developed to study flexoelectricity in (Ba0.5Sr0.5)TiO3 thin films of different thicknesses in their paraelectric phase. The magnitude as well as sign of individual components of the flexoelectric tensor are reported, which provides answers to existing controversies. The use of this numerical tool also allows us to show that flexoelectric coefficients depend strongly on the film's thickness and temperature. Such dependence is explained using the relationship between the flexoelectric coefficients and the dielectric susceptibility.