The third volume of Peter Johnstone's massive compendium of topos theory, "Sketches of an Elephant", is yet to be published. The volume is supposed to discuss cohomology and mathematical universes in the context of topos theory. While we wait for the publication of this volume, are there any alternative (hopefully just as comprehensive) references for the aforementioned subjects?
As I said in the comment, this would involve a very large number of different references! (almost one by subsection...)
But to some extent, contributors to the nLab already started doing that and it is probably the best place to start if you are interested in material covered by this third volume: https://ncatlab.org/nlab/show/Elephant
On this page you have the announced table of content of Sketches of an elephant including volume 3, with most subsection references to page on the nLab covering the idea suggested by the title of the subsection. The nLab page itself will generally not contains enough information to reconstruct the content of the subsection but will surely contains references to other papers and books.