Where is the temporary directory in Linux?

The Filesystem Hierarchy Standard version 3.0 says:

/tmp : Temporary files

The /tmp directory must be made available for programs that require temporary files.

Programs must not assume that any files or directories in /tmp are preserved between invocations of the program.

Rationale

IEEE standard POSIX.1-2008 lists requirements similar to the above section. Although data stored in /tmp may be deleted in a site-specific manner, it is recommended that files and directories located in /tmp be deleted whenever the system is booted.

FHS added this recommendation on the basis of historical precedent and common practice, but did not make it a requirement because system administration is not within the scope of this standard.

/var/tmp : Temporary files preserved between system reboots

The /var/tmp directory is made available for programs that require temporary files or directories that are preserved between system reboots. Therefore, data stored in /var/tmp is more persistent than data in /tmp.

Files and directories located in /var/tmp must not be deleted when the system is booted. Although data stored in /var/tmp is typically deleted in a site-specific manner, it is recommended that deletions occur at a less frequent interval than /tmp.

Also the The Open Group Base Specifications Issue 7, Environment Variables mentions the following:

TMPDIR This variable shall represent a pathname of a directory made available for programs that need a place to create temporary files.

This is an old question so today there is another option available. Linux distributions relying on systemd (which is 90% of them) can now use $XDG_RUNTIME_DIR directory (XDG Base Directory Specification) to store certain types of temporary files. It is generally located at /run/user/$uid. This is a per-user directory with 700 permissions which provides better security. This is a tmpfs mount which provides performance. The downside of tmpfs is that it should only be used to keep small files and sockets.

I look at it as a marriage of /tmp and /var/run.

Yes /tmp is for general use. See here and here On the Filesystem Hierarchy Standard.

/tmp/ Temporary files (see also /var/tmp). Often not preserved between system reboots.

With some more details listed in the PDF.