MD80 Captain reports being dispatched with one generator MEL'd and the APU acting as a replacement power source. During their climb; the APU failed and the flight returned to departure airport.

Narrative

After first push out; left generator failed on engine start. Returned to gate and Maintenance deferred left generator in accordance with MEL; and accomplished required APU MEL (M) checks and installed placards limiting flight to FL350. Flight re-released with new fuel; load; new papers pulled; departed. CSD (left) disconnected after left engine start in accordance with MEL and mechanics brief. On climbout between FL270 and FL320; APU became erratic then failed. Descent initiated to FL300 to attempt restart (older MD-80). Restart attempt unsuccessful. Applicable portions of QRH completed; Captain elected to return to departure airport. Weather [was] 95% VMC. Emergency declared with ATC for overweight landing; Dispatch notified and normal approach and soft landing accomplished; followed by normal taxi in. Logbook completed for overweight landing (8;500 LBS over maximum landing weight limit) and APU failure. After deplaning a retired Captain said he did not hear any of my long winded explanation as to why we were returning (regarding the 2 out of 3 rule regarding electrical power sources and the necessity of our return) over the PA. When I questioned my Flight Attendant crew; one said she heard my explanation while the other two said they did not. All of them heard the required 'prepare' calls; however; I found this odd as I specifically looked at my blue PA light and my meter deflection while I explained the situation. Re-equipped and re-departed. My suggestion is to not dispatch these older aircraft with only one main engine generator and an APU generator. These older APU's are extremely unreliable at altitude in my experience.