Do I really need lithium batteries? (1.5 V, AA)
The page in German appears to NOT say what you claim.
Ich spreche ungefähr null deutsch :-(.
Google Übersetzer sagt:  


Use standard alkaline or lithium batteries for the outdoor sensor in cold weather. DO NOT use high performance batteries or rechargeable batteries.


And a User comment says


The outdoor sensor requires for operation 2 AA batteries, preferably lithium variants, as they work better than the alkali variants in cold weather.  ...


Lithium-Ion and "Lithium" are NOT the same.  

Lithium primary (non-rechargeable) cells are available in a number of chemistries and the ones they mention are a direct replacement for AA-Alkaline ones. They have a similar terminal voltage, higher energy content, flatter discharge curve, better low temperature operation and, usually, a much higher proce than Alkaline cells.
They are useful in niche applications where their special characteristics may be of value or ecen essential, but in most cases AA Alkaline cells work almost as well and are much better value for money.

In the case of your weather station and given the likely minimum temperature range I'd expect the Alkaline cells to be entirely acceptable. Usually.
eg after a long period of use if there was a much colder than usual day the Lithium cells may still operate whjen the Alkaline one would not.
But using Alkaline and replacing periodically will probably be cheaper overall and just as good.



Example of performance of Lithium AA cell.
Key areas noted in red or green.  

Datasheet Energiser L91 Lithium
Similar

17 page datasheet &amp; application manual here


https://data.energizer.com/pdfs/l91.pdf