Let’s say that we have five territories of equal size on a planet far, far away.
With temperatures of 84, 49, 22, 16 and 8 K. It’s the warmest in the early afternoon area and the coldest around the early morning area. 84 and 8 respectively. The average temperature is 35.8 K.
Now, we spin the planet a bit faster, so that the maximal temperature is only 2 K lower, because it can’t warm that much during the shorter day.
We now have 82, 51, 38, 34, 26 Kelvins. As you can see, the lowest temperature is higher, since the night is shorter as well. The average temperature is now 46.2 – more than 10 Kelvins more.
The beauty of this is that the sum of all five temperatures to the power of 4 is 55,855,825 in both cases. Just as Stefan’s law demands it be if we want to have the same energy output, that is to say if you want to satisfy the energy conservation law.
The numbers here are just an example to help you understand the situation. It would make no difference if we had to calculate with a billion areas instead of 5. The same principle applies; when you lower the maximal temperature, you have to increase the other temperatures by more than that, so as to preserve the energy flow. By doing so, the global average temperature goes up, the average is linear but the radiation isn’t.
It’s quite elementary, in fact.