No, leaving the door of an unplugged refrigerator open will not make it cool down faster. In fact, it will have the opposite effect.
When the door is left open, warm air from the surrounding environment enters the refrigerator, which raises the temperature inside. This occurs because the refrigerator's insulation is designed to keep cold air inside and warm air outside when the door is closed. However, with the door open, the insulation is ineffective, and the warm air quickly fills the refrigerator.
The cooling process of a refrigerator, whether plugged in or not, is dependent on the circulation of air through the cooling coils and the refrigeration system's mechanisms. Simply leaving the door open does not provide any cooling effect.
If you need to cool down food or beverages quickly, it's better to keep the refrigerator door closed and consider using other methods such as placing the items in the freezer or using ice packs.