We present the synthesis and physical properties of a new breathing pyrochlore magnet CuAlCr4S8 via synchrotron x-ray diffraction (XRD), magnetization under ambient and applied hydrostatic pressure, heat capacity, and muon spin relaxation/rotation (µSR) measurements. CuAlCr4S8 exhibits positive thermal expansion with concave upward temperature dependence. We observed a sharp antiferromagnetic ordering transition of a purely magnetic nature at 20 K, which shifts by as much as 3.2 K on application of 600 MPa pressure. The breathing factor (B f = J /J) in breathing pyrochlores can be an important parameter to tune the magnetic ground states of the pyrochlore lattice. The breathing factor can be modulated through breathing ratio, the ratio of sizes of the two tetrahedra, by using different elements at A and A' sites in the breathing pyrochlore structure. We find that CuAlCr4S8 has a breathing ratio of 1.0663(8), which is comparable to other sulfur breathing pyrochlores.