One hook and no global variable version of [https://gist.github.com/nciske/6117419]

slug_changed_tax.php

<?php
add_action( 'edited_terms', 'example_terms_edited' );
 
function example_terms_edited( $term_id ){
  
	$tax = 'post_tag'; //category, custom_taxonomy, etc.
  
	global $wpdb;

	$term = get_term( $term_id, $tax );
	$old_slug = $term->slug;
 
	$new_slug = $wpdb->get_var( $wpdb->prepare( "SELECT slug FROM $wpdb->terms WHERE term_id = %d", $term_id ) );
 
	if( $new_slug != $old_slug )
		wp_mail( get_option('admin_email'), 'Slug changed for term #'.$term_id, "Old Slug: {$old_slug}\r\nNew Slug: {$new_slug}" );
	
}

interesting! the only thing I'd add is to place some comments...
cause it is not obvious that this code is taking advantage of an exploit that the slug change doesn’t immediately update the object cache, so you can compare the cached version to the database.