Does the Antonov An-225 have an Auxiliary Power Unit (APU)?
No. It doesn't have an APU — it has two of them.

They are located close to the main landing gears. From An-225 Mriya is the world’s largest aircraft (English version): 


  Auxiliary power plant consisting of two TA-12 turbofans installed in the left and right chassis fairings provides independent power to all systems and starts the engines.


I think "turbofans" is not the proper term here. Actually the TA-12 is (just) a gas turbine (Wiki (Russian), Google translation).

The location of the APUs being much closer to the tarmac might be a reason why Copenhagen airport has (or had) special regulations for its operation on an An-225 (source):


  Special regulations for AN124 and AN225.
  Aeroplane type AN124 and AN225 must not operate their APU unless the pavement
  underneath is protected against heat and blast effects from the exhaust.


I couldn't find a better picture but the exhaust pipe is quite visible here (red circle):


(Source)

I'm not perfectly sure whether the green circle marks the intake, but I guess so.