Climate Clock uses Apple push notifications to update your app icon's badge. Follow these steps to get it working.
Note: if you tried these steps around the first week of March (Feb 28 - Mar 6) and had trouble, please try step 1 again. My server had a couple of issues due to overwhelming demand. Thanks for your patience!
Please note that Apple push notification technology does not let us display negative temperatures below 0. If the current temperature is 0 or below, no badge will be displayed.
In Climate Clock, keep swiping right until you reach the left-most panel (past your first location). This is the Settings panel. Under Icon Badge Temperature, select no, and then yes.
Go to your devices' Settings -> Notifications -> Climate Clock.
Make sure all your Notification settings are ON, including your "Badge App Icon" setting.
After making sure all your settings are ON, go to back to the main Settings screen and turn Airplane mode ON. Wait 30 seconds, and then turn it OFF.
Your icon badge temperature should now be working! If not, proceed to step 3:
Make sure your network isn't blocking port 5223. Port 5223 is needed for Apple push notifications. For more information, look at Apple's Support Page.
Try changing your WiFi DNS to "8.8.8.8, 8.8.4.4" by going to Settings -> Wi-Fi. Then select the network you are currently connected to. Select DNS and enter "8.8.8.8, 8.8.4.4" without the quotes.
After doing the above, completely turn off your device, wait one minute, and then turn it on again.
If you're still having trouble, move on to step 4.
Delete Climate Clock from your device. Go back to the App Store and download the app again - it will be free this time. (You will have to add your locations and turn Icon Badge Temperature on again in Climate Clock's Settings.)
In rare cases, your device will need to be backed up and restored. Follow Apple's Support Page on restoring your device.
However if you're having problem with your jailbroken device, here's a hint: try searching for "pushfix" - that can sometimes help!