Had this problem as well. This was NOT caused by an upgrade or any other change; this is 100% caused by the maintenance mode plugin's back end. I neither upgraded nor made any other change to the site configuration, but my users started reporting problems about 8 hours ago (the same time this thread started). Setting the 'support' option to false didn't work for me, I had to disable the plugin entirely.
The culprit is the
lrss_check_update() function in key-check.php (see MadMakz's post above). New Relic monitor confirmed that in every case lrss_check_update hangs and times out after 120 seconds (that being the default for my particular configuration):
Component Total(ms) % Max(ms) Call count
lrss_check_update 120,000 99.8% 120,000 1
Internal fun... 41.1 0.0341% 41.1 1
apply_filters 15.1 0.0125% 0.15 3,490
wpdb::query 12.5 0.0104% 1.45 32
get_option 6.66 0.0055% 0.04 339
translate 5.67 0.0047% 0.02 1,010
WTF??? I won't be using this plugin again. Ever.