How to determine whether a PostgreSQL Hot Standby is fully mirrored?
I think this is normal and expected if your restore_command is set to something like this example:

restore_command = 'cp /mnt/server/archivedir/%f "%p"'


The manual says that:


  At startup, the standby begins by restoring all WAL available in the
  archive location, calling restore_command. Once it reaches the end of
  WAL available there and restore_command fails, it tries to restore any
  WAL available in the pg_xlog directory. If that fails, and streaming
  replication has been configured, the standby tries to connect to the
  primary server and start streaming WAL from the last valid record
  found in archive or pg_xlog. If that fails or streaming replication is
  not configured, or if the connection is later disconnected, the
  standby goes back to step 1 and tries to restore the file from the
  archive again. This loop of retries from the archive, pg_xlog, and via
  streaming replication goes on until the server is stopped or failover
  is triggered by a trigger file.


So you can expect to see exactly one restore_command failure when you start your standby, because PostgreSQL will keep calling it (with incrementing log file names/numbers) until it fails once.

Then it will connect to the primary and start streaming as described above, and as you saw in your logs:

LOG:  streaming replication successfully connected to primary


The slave is not guaranteed to be exactly up-to-date with the master, because it could be disconnected from the master for example. In particular, this line:

LOG:  consistent recovery state reached at 31/B73624A0


does not mean that "the hot standby contains all the data of the master". However, if you see it followed by this line, as you did:

LOG:  database system is ready to accept read only connections


then the database is "ready enough" to start functioning as a read-only standby, as the manual says:


  It may take some time for Hot Standby connections to be allowed,
  because the server will not accept connections until it has completed
  sufficient recovery to provide a consistent state against which
  queries can run. During this period, clients that attempt to connect
  will be refused with an error message.


In my case, I saw consistent recovery state reached not followed by database system is ready to accept read only connections. This turned out to be a problem with an embedded scripting language plugin (plpython2) having a system-wide startup script (sitecustomize.py) which did bad things to the PostgreSQL process (enabling faulthandler and installing a signal handler for SIGUSR2) which caused it to never enter hot standby mode.