Why are the center tank pumps deactivated when slats are extended on the A320?
You're right. But it's not because of the slats. It's because of what the plane is doing when the slats are being used.

The slat position is used in some system logic because it's indicative of a takeoff and approach. For example, on the MD-11 the engine ignition is automatically put into continuous mode when the slats are operated, because that's indicative of slow speed.

The same for the A320, slats extended means a takeoff / climb is about to commence, or the plane is on approach with a potential go-around.

Those conditions require high fuel pressure.

The center tank has 2 pumps, while each wing tank has 2 pumps. If a center tank pump fails, it will be 1 pump supplying 2 engines, compared to 2 pumps for each engine (or 1 for each in a rare double pump failure—one in each wing).

If a pump fails, there is no automatic command that can be sent to the sequence valves to change the sequence, or to engage the crossfeed valves.*

Usually the center pumps overpower the wing pumps, but what is omitted from that explanation is that there are sequence valves. If one center pump fails, the pressure may remain high enough to keep the valves closed on the good side, but not high enough to supply two engines during high demand.

Also note there is no suction (gravity) feeding from the center tank. I attached an A320 fuel schematic below, number 13 is the sequence valve.



RE: Why does not this apply to all aircraft?

The 737 Classic faced issues (see footnote). Other types have more than 2 pumps for the center tank, and/or have more powerful pumps. Another option is to use the center tank to fill the other tanks (like on the MD-11).

The 747-1/200 also required using the wing tanks during takeoff, see this answer by @Terry.

A speculation on my part, is that it was cheaper to certify the plane like that, or problems were found and that was the cheapest solution.


  A fuel tank that is not designed to feed the engines under all flight conditions need
  be tested only for the flight regime for which it is designed to do so (e.g., cruise conditions).
  
  (FAA Flight Test Guide For Certification
  Of Transport Category Airplanes)


* A fuel imbalance may quickly develop when each engine is consuming +70 kg of fuel per minute (an issue that affects the 737 Classic).


  On the classics, when departing with less than 1,000kg of fuel in the centre tank, an imbalance may occur during the climb. This is because the RH centre tank pump will stop feeding due to the body angle so number 2 engine fuel is drawn from main tank 2, while engine 1 is still drawing fuel from the centre tank. When this “runs dry” the scavenge pump will also transfer any remaining centre tank fuel into main tank 1, thereby exacerbating the imbalance.


(A320 FCOM)