What does "database.suspend.dat" gateway files used for?

This file gets written and removed seamlessly in the background, but in the event that you become aware of such a file, it is most likely due to gateway log messages such as these:

INFO: RoleManager Receiving transfer of non-existent file 'file://./database/database.suspend.dat'
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WARN: RoleManager Failed to backup 'file://./database/database.suspend.dat' to 'hotStandby_backup_20141122_140836/./database/database.suspend.dat'.
Error File access error. File "./database/database.suspend.dat". Error (2), No such file or directory
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The error is relating to a file called database.suspend.dat - this file is written when database logging in effect is suspended - as you have hot standby activated this file gets copied over when created / modified to your secondary gateway server. The messages above in this respect is that the file could not be copied over indicating that (1)  the file might already been deleted or removed on the primary server hence the INFO message; or (2) there does not seem to have enough file/directory permissions on the secondary gateway server to facilitate the copying, hence the WARN message.

These messages will only be apparent when you've had Database Logging suspended previously, otherwise you should not be encountering these errors on normal operations.