FFT Frequency Bin - Impact on Energy Totals
One of the most useful ways of handling this sort of question with FFTs is to use Parseval's Theorem. Paraphrasing loosely, it says that if you have a signal, it doesn't matter whether you compute its energy by summing the power in each time sample, or the energy in each frequency bin, you necessarily get the same answer - because it's the same signal.
If you keep the sample rate the same, and halve the number of samples you take, then you've halved the length and so halved the total energy in the signal.
Halving the number of samples will halve the number of frequency bins, while doubling their width.
Half the total energy, half the number of frequency bins, therefore the energy in each bin stays the same.