The nations with the highest total fertility rates (Niger, Chad, Somalia, DRC...) all have far lower rates of energy consumption per capita than sub-replacement-fertility nations like South Korea, Italy, Japan, or Canada. Energy availability isn't a strong proxy for or constraint on fertility rates, at least in the next few decades.
One of the most important needs to enable fertility is a temperature in which an infant can survive. In colder climates, that requires a lot more fuel input to heat homes, whereas in consistently warm climates, like the African countries to mentioned, they get their heat "for free", and therefore fertility is less constrained.
The heat poses other problems that affect survival rates, like tropical diseases, but the energy cost of fighting those is far less than heating homes.
The same observation applies to the agricultural sectors of those countries, which can be incredibly productive due to growing regions with consistently warm weather. It's how China and India can support such massive populations (both are largely food self sufficient). And it's also why Brazil can produce sugar cane based ethanol far more efficiently than the US can produce corn based ethanol.
Once this major difference is accounted for, however, change in energy consumption over time is a reasonable proxy for change in standard of living.
But absolute differences in per capita energy consumption fail to capture variations due to things like country size, transportation/energy infrastructure, and socioeconomic structure i.e Russia has a much higher per capita energy consumption than Germany, but a lower standard of living.
