Picking battery for power amplifier
Here are some guidelines. Look at the data sheet to determine: -


The battery voltage needed to achieve 20 watts bridged output
The speaker impedance to achieve that power out
The peak current that can be expected at this power
Quiescent current when not driving sound to the speakers


Then, start to narrow down your choice of battery - battery voltage should become apparent and peak handing current for the battery also. You also need to decide the duty cycle of producing power to the quiescent time (when the amp is only consuming low current) to determine the ampere-hours of the battery over a typical period of time between re-charges. Get this number and double it (playing safe)

This dictates the capacity of the battery.

Regards efficiency - it's a class AB amp and the data sheet should tell you the efficiency.