Calculating life of portable power supply for icybreeze cooler
You have fallen prey to confusing specmanship.

Start with the cooler. The data sheet says that its battery is 12 volts 10 Ahr. This is 120 Whr. At high power, this will last 2.5 hours.

Now, about your power source. Yes, its battery is rated for 42 Ahr. However, that is produced by a 3.7 volt (nominal) Li-ion battery. Note that the data sheet says the total energy is 146 Whr. This is about 20% higher than the cooler's battery, so figure on an endurance (from the power source only) of 1.2 x 2.5, or 3 hours.

So, the cooler will run about 2.5 hours on its own battery, and the power source will run an extra 3 hours, for a total of about 5.5 hours. Does that sound familiar?

Please note that your power source has a 120 volt, 1 amp AC output. Since this draws 120 watts, at that level the power source will last for 146/120, or about 1.2 hours.

The key is that 120 volts (the output) is about 32 times greater than 3.7 volts (the battery voltage). So, in order to keep the power equal, a 1 amp output at 120 volts will require about 32 amps from the battery, and your 42 Ahr battery is not going to last very long.