  We present the synthesis and physical properties of a new breathing
pyrochlore magnet CuAlCr$_4$S$_8$ with the help of synchrotron x-ray
diffraction (XRD), magnetization under ambient and applied hydrostatic
pressure, heat capacity, and muon spin relaxation/rotation ($\mu$SR)
measurements. CuAlCr$_4$S$_8$ 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 the 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 CuAlCr$_4$S$_8$ has a breathing ratio of 1.0663(8), which is
comparable to other sulfur breathing pyrochlores.
