Rechargeable battery calculation
First, stating the background based on the device now shown in the question:

The emergency charger shown is designed to provide 5 Volts at 500 mA, from 2 AA cells.

It is not clear from the device photo alone, and without specifications being provided, whether the device uses the batteries in series (will not work with a single cell inserted), or parallel as the question implies (will work even with a single cell inserted). 

The circuit inside would be a boost converter, generating 5 Volts from a 1 to 3 Volt input. Typical boost converter efficiency is anywhere from 75 to 90 percent, depending on design and component selection. 

Assumptions:


85% efficiency of boost converter for calculations
Eneloop HR-3UTGA NiMH rechargable AA cells
The device uses the AA cells in parallel (series calculations also follow)




Output power: 500 mA @ 5 Volts = V x I = 2.5 Watts
Required input power:  2.5 / 0.85 = 2.941 Watts (at 85% efficiency)
Required min current: 2.45 Amperes (P / V, fully charged NiMh 1.2 Volt battery)
Required max current: 2.94 Amperes (with depleted battery, at 1 Volt)


Assume roughly equal split of current draw...

Min current from each: 1.225 Amperes
Max current from each: 1.47 Amperes
Actual capacity: ~ 1800 mAh per AA (Discharge graph in datasheet, between 1 and 2 A)
Capacity of 2xAA: ~ 3600 mAh
Time to Discharge: Capacity / Input Power = 3600 / 2941 =  ~1.224 hours


So, even in optimistic conditions, the charger is expected to deliver less than 1 hour 14 minutes of power, from the above calculations. In reality, the current drawn would rise as voltage drops, thus causing faster depletion. Also, whichever cell has lower internal resistance due to manufacturing and life-cycle differences, would discharge first, loading the second into a quicker depletion.

Thus the observed 1 hour of operation is not unexpected.



Now, if the batteries were connected in series, as is possible, the following calculations change, the rest remain as above:

Required min current: 1.226 Amperes (fully charged, 2 x 1.2 = 2.4 Volts)
Required max current: 1.47 Amperes (with depleted battery, 2 x 1 Volt)
Actual capacity: ~ 1800 mAh per AA (Discharge graph in datasheet, between 1 and 2 A)
Capacity of 2xAA: ~ 3600 mAh
Time to Discharge: Capacity / Input Power = 3600 / 2941 =  ~1.224 hours


Thus, operation time would be about the same, but single battery operation would not be an option.



Additional notes:


Typical boost converter efficiency reduces with increase in gap between input and output voltage. This impacts both faster depletion towards the end of battery charge, and lower efficiency for a parallel battery arrangement.
Thus, realistically the parallel battery option is likely to last much less than with batteries in series.
The probable reason the device heats up more with rechargeable cells than with standard 1.5 Volt cells, is that the lower voltage causes greater efficiency losses. Such losses are typically dissipated as heat in electronic devices.