Reading through old Techlink Articles I found this gem:
A differential cooler assembly is available for the Cadillac STS-V (fig. 1). The part number is 15868857.
The system is designed to reduce the operating temperature of the rear differential fluid when the STS-V is driven at extended high speed or in competitive driving. The STS-V comes with control logic to operate the system when high heat generation conditions are met. The wiring harness has provisions for the cooler, making it an easy installation. Installation time is less than 2 hours.
The differential lubricant pump and cooler assembly is located under the vehicle and is mounted to the rear storage compartment well. The cooler module assembly consists of a pump, a heat exchanger, an electric fan and assorted hoses/fittings. All components except the external lines and fittings are located in the Cooler Module.
Oil is pulled from the differential and pumped through a hose to the cooler/heat exchanger. The differential oil is cooled by dissipating heat as the 12 volt electric fan blows outside air over the heat exchanger. The cooled differential oil then returns to the differential.
TIP: The pump and electric fan can be turned on using the Tech 2 or suitable scan tool.
The differential cooler system has a maintenance feature that operates the system for approximately 15 seconds under certain conditions during initial vehicle operation. It is normal to hear the system run during this maintenance cycle.
If the vehicle is equipped with this system, rear differential fluid capacity is an additional 0.31 L (0.23 qt).