Login into your server as “root” and start firing the below commands,
# cd /usr/local/src
# mkdir mod_pagespeed
# cd mod_pagespeed
# wget https://dl-ssl.google.com/dl/linux/direct/mod-pagespeed-stable_current_x86_64.rpm
# rpm2cpio mod-pagespeed-stable_current_x86_64.rpm | cpio -idmv
# cp /usr/local/src/mod_pagespeed/usr/lib64/httpd/modules/mod_pagespeed.so /usr/local/apache/modules/
# cp /usr/local/src/mod_pagespeed/etc/httpd/conf.d/pagespeed.conf /usr/local/apache/conf/
# chmod 755 /usr/local/apache/modules/mod_pagespeed.so
# mkdir /var/mod_pagespeed/{cache,files} -p
# chown nobody:nobody /var/mod_pagespeed/*
mod_pagespeed has a dependency that you’ll want to enable: mod_deflate (the httpd source directory (httpd-2.2.21) may vary depending on your install):
#/usr/local/apache/bin/apxs -c -i /home/cpeasyapache/src/httpd-2.2.21/modules/filters/mod_deflate.c
After that, we’ll have to edit the mod_pagespeed configuration file located at /usr/local/apache/conf/pagespeed.conf to reflect the correct paths, the LoadModule directives should fetch the modules in “modules/module_name.so”:
LoadModule pagespeed_module /usr/local/apache/modules/mod_pagespeed.so
LoadModule deflate_module /usr/local/apache/modules/mod_deflate.so
ModPagespeedFileCachePath “/var/mod_pagespeed/cache/”
ModPagespeedGeneratedFilePrefix “/var/mod_pagespeed/files/”
And then open /usr/local/apache/conf/includes/pre_main_global.conf and add:
Include “/usr/local/apache/conf/pagespeed.conf”
Next restart apache
# /scripts/buildhttpdconf
# /etc/init.d/httpd restart And we are done!

See more at: http://datlinux.blogspot.my/2013/06/how-to-install-modpagespeed-in-cpanel.html