Where did you get this info? Not to be contentious, but AH rating is AH rating. Discharging a lithium battery too far can cause issues. Trust me, I design battery solutions for military and space. The battery should shut down to prevent discharging too far, since this can be dangerous, as in it may catch fire and explode when you recharge. That is why every battery pack carries an onboard computer that monitors the charge state and is supposed to shut down if it is charged too fast, overcharged too far or discharged too far.
I will look into these battery packs and get back with further "FACTS"
Back with facts... This unit will give you that number of AHs before protecting itself. 40AH is not that much capacity and is based upon the battery voltage, not 120V. So it will not give you 40 amps at 120v for an hour.
If you read the fine print, it is 444 watt hours. If your cooler pulls a constant 3 amps at 12 volts you should be able to run for approximately 444/(3x12)= 12.3 hours under ideal conditions. Now you have to figure in the duty cycle of the fridge, in other words how much of the time is it actually "on". If it is actually providing cooling one third of the time, then you should be able to run around 37 hours if running from the 12v outlet. If running from the 120v volt outlet, it will be less due to the losses in the conversion process. I would estimate the losses to be around 20% for a well designed inverter, with 90% being exceptional.
While it is a very nice, small, and handy unit, it is not the answer to everybody's needs. For what it costs, if you have the space, you could put an additional dual deep cycle battery setup in your rig and have several times more capacity, and add an inverter for the 120v.