You can create your custom pagination using paginate_link()
function and add custom query string after current url.
To add argument to the links of paginations you can pass them as array argument inside the ‘add_args’
if ( is_home() || is_archive() || is_search() ) :
echo paginate_links(array(
'base' => preg_replace('/\?.*/', "https://wordpress.stackexchange.com/", get_pagenum_link(1)) . '%_%',
'current' => max(1, get_query_var('paged')),
'format' => 'page/%#%',
'total' => $wp_query->max_num_pages,
// here you can pass custom query string to the pagination url
'add_args' => array(
'sort_by_type' => ( !empty($_GET['sort_by_type']) ) ? $_GET['sort_by_type'] : 'vote'
)
));
endif;
Replace your pagination code with this above code.
It will print all string with custom query string.
http://example.com/page/2/?sort_by_type=vote
Hope this help!