Seems to be a French word equivalent to atmosphere. I guess it just never caught on.
In Google n-gram viewer "atmosphere" is shown to occur in 0.003% of printed materials since year 1800, as far back as the graph goes.
"Aerosphere" did not show up until 1850, has averaged around 0.00000012% since then, with a big uptick around year 2000 to 0.00000022%.
There is one use of the word "aerosphere" for every 14,000 uses of "atmosphere".