Can Apex static values be used to safely communicate between executions of process builder?

Process Builder is bulkified, which means it would only perform one query per 200 records. That said, if you wanted to store data in a static variable, it would persist across the entire transaction, in before triggers, after triggers, and yes, even InvocableMethod calls. I would suggest simply trying the query-then-assign method by using Process Builder + Flow unless you really need the extra performance.

If your Process Builder is doing any record field updates then be aware that each of these is dealt with in a separate invocation of your entire "save procedure", as per point 14 in the order of execution documentation, but as part of the same transaction/session.

This can cause you all sorts of trouble if any related triggers are not written to cache relevant query results (e.g. easily exceeding the SOQL number of queries limit, the SOQL number of records limit and even the CPU limits). Take a look at the governor limits documentation for further info on limits you face.

Of course, the queries you need to cache may actually vary on the records input to the process, and maybe even the data in those records, so you might have to do some clever stuff to invalidate parts of the cache and requery some data if inputs to the query change (due to specific changes in the records). This may then mean you are scuppered anyway (e.g. because each iteration through the PB/trigger sequence causes additional changes that invalidate the cache) and here you then have to refactor the solution, likely removing the PB and replacing it with processing in your trigger(s).