Index: src/wp-includes/class-wp-feed-cache.php
===================================================================
--- src/wp-includes/class-wp-feed-cache.php	(revision 49560)
+++ src/wp-includes/class-wp-feed-cache.php	(working copy)
@@ -5,8 +5,16 @@
  * @package WordPress
  * @subpackage Feed
  * @since 4.7.0
+ * @deprecated 5.6.0
  */
 
+_deprecated_file(
+	basename( __FILE__ ),
+	'5.6.0',
+	'',
+	__( 'This file is only loaded for backward compatibility with SimplePie 1.2.x. Please consider switching to a recent SimplePie version.' )
+);
+
 /**
  * Core class used to implement a feed cache.
  *
Index: src/wp-includes/feed.php
===================================================================
--- src/wp-includes/feed.php	(revision 49560)
+++ src/wp-includes/feed.php	(working copy)
@@ -789,7 +789,6 @@
 		require_once ABSPATH . WPINC . '/class-simplepie.php';
 	}
 
-	require_once ABSPATH . WPINC . '/class-wp-feed-cache.php';
 	require_once ABSPATH . WPINC . '/class-wp-feed-cache-transient.php';
 	require_once ABSPATH . WPINC . '/class-wp-simplepie-file.php';
 	require_once ABSPATH . WPINC . '/class-wp-simplepie-sanitize-kses.php';
@@ -801,7 +800,16 @@
 	// constructor sets it before we have a chance to set the sanitization class.
 	$feed->sanitize = new WP_SimplePie_Sanitize_KSES();
 
-	$feed->set_cache_class( 'WP_Feed_Cache' );
+	// Register the cache handler using the recommended method for SimplePie 1.3 or later.
+	if ( method_exists( 'SimplePie_Cache', 'register' ) ) {
+		SimplePie_Cache::register( 'wp_transient', 'WP_Feed_Cache_Transient' );
+		$feed->set_cache_location( 'wp_transient' );
+	} else {
+		// Back-compat for SimplePie 1.2.x.
+		require_once ABSPATH . WPINC . '/class-wp-feed-cache.php';
+		$feed->set_cache_class( 'WP_Feed_Cache' );
+	}
+
 	$feed->set_file_class( 'WP_SimplePie_File' );
 
 	$feed->set_feed_url( $url );