How much more energy is required to maintain a higher temperature in one of two otherwise identical containers?

What you need to know is not the energy but the amount of power, which is the rate of energy transfer per unit time. You need to be talking Watts rather than Joules.

It depends on how well insulated the containers are, the temperature of the surroundings and the shape and surface area of the containers. The content of the containers is only an issue when you are heating the containers, not when you are maintaining the temperature of the containers.

<<How much more energy is required to maintain a higher temperature in one of two otherwise identical containers?>>

Obviously it depends on the specific temperature difference you want.

If the containers are in vacuums so the only loss is from blackbody radiation the amount of power needed goes as the temperature to the 4th power (temperature measured in absolute units like Kelvins), so, for example if you wanted one container to be at a temperature 3 times as large as the other, the power given to it at steady state must be 3 to the 4th power as much, which is 81.

However, blackbody radiation is not the only way a body can lose energy. There are processes where the loss is proportional to the temperature difference between the body and the environment in thermal contact with it. So, for example, if the environment was at 50 degrees and one body was at 60 and the other was at 90, since 90-50 is 4 times as great as 60-50, 4 times as much power would need to be pumped into the hotter one.

For thinks happening on Earth, the second process is dominant. But for global warming, since "space" is largely a vacuum (and thus there is no "thermal contact") blackbody radiation is the operative process.

I rarely rag on someone for category misplacement but this belongs in physics.

OK that said - You are looking at the thermal equivalent of a Voltage - Resistance type problem.

The container likely has a fairly linear Temperature difference to Heat transfer ratio.

Hence a doubling of the delta Temp would likely double the heat (aka energy) leaving the container & consequently the energy required to maintain the higher temperature.

It depends on the stuff in the container, the higher the temperature, the higher the energy level. It is basic thermodynamics.

depends on a lot of things. specific heat of the contents/container, ambient temperature, thermal conductivity of the container plus the overall heat transfer coefficient.