Can I use LSTM models to evaluate multiple, independent time series?
If you are looking to predict multiple time series (which would be similar in nature, since each weather station in the area would record similar temperatures, even if they are not identical), using a separate LSTM model for each may prove quite time-consuming.

One approach you could take is one suggested in an excellent answer for another question on Cross Validated. Essentially, the author is describing a means for forecasting sales with LSTM whereby the model is trained on a mini-batch (or subset) of one series, and then a new series is selected. In this case, I would understand this to mean that a subset of data is incorporated from weather station 1, then another batch from weather station 2, etc.

This would have the advantage of essentially creating a unified series that takes the characteristics of all weather stations into account - which allows for maximum utilisation of the data, as well as allowing the network to learn patterns from all stations - not just one or a select few.

One option is to try this method using a subset of data for select weather stations at first, and then compare the prediction accuracy to those that use separate LSTM models for each. If you find that the predictions for the former method are more accurate, then it would make sense to go with that.

If you are using R, then there is also the option of using a package like forecastML to do so - if you are not tied down to using an LSTM model then this could also be another consideration.