How to maintain a file's 'last modified' date?
cp --preserve=timestamps <source> <destination>
If you are recursively copying you could use:
cp -a <source> <destination>
From the cp manual page:
-a, --archive
same as -dR --preserve=all
-d same as --no-dereference --preserve=links
-R, -r, --recursive
copy directories recursively
--preserve[=ATTR_LIST]
preserve the specified attributes (default: mode,ownership,timestamps),
if possible additional attributes: context, links, xattr, all