Unfortunately, the logging of the build network geoprocessing tool is not cumulative. This is also explained in the documentation: "Build errors are also written to a BuildErrors.txt file in the temp directory. This file only stores the errors from the last build performed; so it is not cumulative." (Fixing the errors reported after building the network—Help | ArcGIS Desktop )
This is not very handy since the build process can sometimes run several hours. Especially ArcGIS Pro promotes chaining or batching geoprocessing tasks with the task pane. As a user I want to be able to run several build processes successively and check the logs sometimes only at the next day. With the current implementation, log files from older build processes are overwritten. It would be more helpful and would support the design of ArcGIS Pro geoprocessing better, if logs are written in subfolders per process, or if log file names are time-stamped.