cp recursive with specific file extension

You can either let the (bash) shell do the recursion instead:

shopt -s globstar
cp -nt ~/copies/ ~/Documents/**/*.odt

or use the find command

find ~/Documents -name '*.odt' -exec cp -nt ~/copies/ {} +

The former may exceed the system's ARG_MAX limits if the number if files is very large, whereas the latter will break the command up to stay within ARG_MAX.

I added the -n option to prevent "clobbering" files with the same name.

If you wish to replicate the directory structure (more like what cp -r does as suggested by steeldriver) but only populate it with the .odt files, you can do it with rsync, where -f is the 'filter' option. The other options are more straight-forward. You find all the options in man rsync.

rsync -nvr -f '+ *.odt' -f '+ **/' -f '- *' --prune-empty-dirs ~/Documents/ ~/copies/

It is a good idea to use the 'dry run' option -n and the verbose option -v in order to see what will be done before running the real command (when n is removed from the command line).