You have a problem with your unit defintions here that must be corrected.
It's not Amps/Hour. It's Amp-Hours. Amps/Hour would mean that the more hours you run the more amps you can pull from the battery before it runs out. If you have Amp-Hours and divide by amps consumed, the "amps" divides out of the equation leaving you with just "hours" the battery will last for which is what you want.
It's similar to how torque is Newton-Meters, not Newtons/meter. Newtons/meter would mean that if you applied torque to a lever arm, more force would appear at the the tip if it was longer, when in fact the force at the tip is less if the arm is longer.
The same thing is true for Watt-Hours. It's Watt-Hours, not hours per watt. A device will
consume X Watts/Hour which is a measure of
power consumption. But a device stores Watt-Hours which is a measure of
energy storage:
Watts * Hours = Joules/Second *Hours = Joules/Second * 3600 Seconds = 3600*Joules
The time units cancel out leaving the final number in Joules- a measure of energy.
Is that clear?
So a 10A/hr battery, load 5A/hr will last 2 hours --- double the voltage delivery, should last 1 hr. (i.e. 10A drawn in the hour).
But: 10W/hr doing 5W of work per hour will last 2 hours -- double the voltage delivery, you square the work done, i.e. 25W (therefore will last 15 minutes)
5 Watts of work per Hour does not make sense. Work is Joules (energy). Watts is power. You could say 3600 Joules of work per hour which is basically saying 1W of power (Power = 3600 Joules/3600 seconds). Power output (Watts) is the same whether or not you output it for 1 second or a million years- what will be different is the total energy.
Power (Watts) = Energy (Joules) * Time (Seconds)
For current draw saying 5A/hour is incorrect (saying it draws 5A for one hour is a different thing). 5A is 5A, regardless of whether it exists for one second or a million years. Just like how Watts is Joules per second, amps is a measure of how much charge which is Couloumbs/second. Saying That Watts Per Second or Joules Per Hour doesn't make sense. You don't say that you are driving at 50km/h per hour, do you?
Do not move on until you understand this because it is part of the problem (and will cause many MANY problems elsewhere).
So a 10A/hr battery, load 5A/hr will last 2 hours --- double the voltage delivery, should last 1 hr. (i.e. 10A drawn in the hour).
But: 10W/hr doing 5W of work per hour will last 2 hours -- double the voltage delivery, you square the work done, i.e. 25W (therefore will last 15 minutes)
What's the problem? They are both measures of battery capacity but not equivelant.
Amps-Hours (not amps/hour) is not a direct measure of energy storage since it only accounts for current (or more accurately stored charge since Voltage*current make power, not energy), not voltage.
Watt-Hour (not watts/hour) is a direct measure of energy storage since it accounts for both Voltage and Current. Furthermore, 5 Watts is not the same thing as 5A so there is no reason that doubling the current draw should have the same effect as doubling the power draw. To convert between the two you NEED to work in that voltage in Amp-Hours, effectively converting it to Watt-Hours.