Online version of All pair shortest path when path weights are updated
I think adding the keywords decremental/incremental might help you find the relevant papers. 

See e.g. 
Camil Demetrescu, Giuseppe F. Italiano: Decremental All-Pairs Shortest Paths. Encyclopedia of Algorithms 2008 for an overview, Camil Demetrescu, Giuseppe F. Italiano: Fully dynamic all pairs shortest paths with real edge weights. J. Comput. Syst. Sci. 72(5): 813-837 (2006), and Camil Demetrescu, Giuseppe F. Italiano: A new approach to dynamic all pairs shortest paths. J. ACM 51(6): 968-992 (2004)

Approximate versions have been considered as well; the most recent result I know of is Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai: Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization, FOCS 2013.
Table 1 in their paper has an overview of results.