The after-run pump comes on after the engine has been shut off but only if the after-run coolant temperature ("thermal") switch F98 (PN 054919369B) located under the intake manifold, as shown below (Item 15 - thermal switch) hits 110 deg C.


Here is the F98 After-run coolant switch in its native habitat:



The switch triggers a relay (No. 324 right kick panel -see photo below) that keeps the pump on for a fixed period, e.g. 10 min.

The pump itself(PN 034 965 561C) is fairly exposed, as shown here and here:



It fails as shown in the photos, i.e. the plastic of the pump gets old and brittle and cracks.
Repairs do not work. Needs to be replaced. Inlet and outlet hoses not not seem to fail so I wouldn't worry about them.

Replacement is easy. Clamp off the inlet and outlet hoses away from the pump, either using hose pinch clamps or needle nose vice grips (don't over tighten). Then remove the hose gear clamps and work the hose off of the pump. (You will only lose a bit of coolant). Remove the electrical connection and remove the pump. Replace with a 3/4" plastic 90 deg plastic hot water elbow (or a copper one if you have one) for roadside (or temporary) repair OR with a new pump. Here is a link to Swordfish's new and improved after-run pump bypass DIY

The early VR6 cars use the same pump as the UrS, the later VR6 cars use a pump with a different electrical connector (i.e. you need to change the wiring). It has been reported that an OE UrS / Early VR6 pump could be purchased at one time from FAP99 (Tim) for $92 + shipping.

A B5 2.7t S4 V6 after run pump (PN 078 965 561) will also work, albeit with minor modifications to the mount,as explained in Zwoobah's AW post:

Link to the B5 pump info

Sean D (quattro20v) came up with a slick trick to make the pump come on sooner. CLICK HERE

It could also happen that the relay creating the 10 minute run-time fails in the run mode (which will drain the battery when the engine is off). Here is the location of this relay:


Photo courtesy of SJM Autotechnik

The PN for that "324" relay is 447965571A. They fail. Get stuck and run the fan and pump until your battery is dead.

This relay is common to many turbo Audis (100s, 200s, UrS4/6, S2, RS2, A4 and even the RS4) as well as some 1998 and up Passats (presumably 1.8t cars) (still labled "324" and in position 3 of a 7 or 8 position relay plate), so you might be able to score one at a wrecking/breaking yard.

Hope that helps.