When I looked up the word pastime, it is defined as "an activity for pleasure" by Cambridge Dictionaries. Therefore, it already include a meaning of "activity." Besides that, Longman Dictionaries gives the word hobby as a synonym and there are not the word "activity" in its example sentences. However, I can see that in Google Ngram, these two words are used jointly. What do you think? Is it grammatically correct or is it non-standard English of writing?
You are correct in your observation that the definition of "pastime" already includes the definition of "activity", hence using "pastime activity" is redundant. This type of phrase is called a pleonasm and is common when speakers aren't aware of the correct meaning of the word, or simply by mistake. More common pleonasms include "CD disc" or "LED diode" (caused by unawareness of the abbreviation).
If used as a mistake (like in this case), pleonasm is a stylistic error. If you want to use it as a literary tool (to express natural speech for example), it is fine, but otherwise using simply "pastime" is more correct.