Global warming is one of most severe environmental concerns that our planet is facing today. One of its causes is the previous generation of refrigerants that, upon release, remain in the atmosphere for longer periods and contribute towards global warming. This issue could potentially be solved by replacing the previous generation’s high global warming potential (GWP) refrigerants with environmentally friendly refrigerants. This scenario requires an analysis of new refrigerants for a comparison of the thermodynamic properties of the previously used refrigerants. In the present research, a numerical study was conducted to analyze the thermodynamic performance of specifically low GWP hydrofluoroolefens (HFO) refrigerants for an actual vapor compression refrigeration cycle (VCRC) with a constant degree of 3 K superheat. The output parameters included the refrigeration effect, compressor work input, the coefficient of performance (COP), and the volumetric refrigeration capacity (VRC), all of which were calculated by varying the condenser pressure from 6 to 12 bars and vapor pressure from 0.7 to 1.9 bars. Results showed that R1234ze(Z) clearly possessed the desired thermodynamic performance. The drop in refrigeration effect for R1234ze(Z) was merely 14.6% less than that of R134a at a 12 bar condenser pressure; this was minimum drop among candidate refrigerants. The drop in the COP was the minimum for R1234ze(Z)—5.1% less than that of R134a at a 9 bar condenser pressure and 4.7% less than that of R134a at a 1.9 bar evaporator pressure, whereas the COP values of the other refrigerants dropped more drastically at higher condenser pressures. R1234ze(Z) possessed favorable thermodynamic characteristics, with a GWP of 7, and it can serve as an alternative refrigerant for refrigeration systems for a sustainable environment.