The thermostatic expansion valve reduces the flow rate as the evaporator:

The thermostatic expansion valve (TXV) functions to manage the amount of refrigerant that enters the evaporator based on its temperature, ensuring optimal performance of the refrigeration system. The correct choice, indicating that the flow rate is reduced as the evaporator cools, reflects how the valve responds to the temperature of the refrigerant leaving the evaporator coils.

When the evaporator cools, the temperature of the refrigerant exiting is lower, which signals the TXV to reduce the flow of refrigerant entering the evaporator. This action helps maintain proper saturation conditions and prevents excessive refrigerant flooding into the evaporator, which can lead to inefficient cooling and potential damage to the system. The TXV's ability to modulate the flow based on the evaporator's cooling status is crucial in maintaining system efficiency and performance.

In contrast, the other options do not accurately depict the operation of the TXV. For instance, if the evaporator were to warm up, it would signify a demand for more refrigerant to enhance cooling, which is the opposite effect. The option relating to the system simply running does not address the critical function of temperature response.