Variations of the leakage current behaviors and interface potential barrier $({Phi}_B)$ of rf-sputter deposited (Ba, Sr)$TiO_3$ (BST) thin films with thicknesses ranging from 20 nm to 150nm are investigated as a function of the thickness and bias voltages. The top and bottom electrodes are dc-sputter-deposited Pt films. ${Phi}_B$ critically depends on the BST film deposition temperature, postannealing atmosphere and time after the annealing. The postannealing under $N_2$ atmosphere results in a high interface potential barrier height and low leakage current. Maintaining the BST capacitor in air for a long time reduces the ${Phi}_B$ from about 2.4 eV to 1.6 eV due to the oxidation. ${Phi}_B$ is not so dependent on the film thickness in this experimental range. The leakage conduction mechanism is very dependent on the BST film thickness; the 20 nm thick film shows tunneling current, 30 and 40 nm thick films show Shottky emission current.