Download and install the latest software for Mountain Lion below:
Be sure to select HP Recommended Software during the installation to get full functionality as OCR and front button scanning.
Once completing the installation reboot your Mac and run Software Update to ensure you have the latest drivers.
Apple only distributes the printer drivers for vendors like HP. Availability, status and functionality including compatibility with specific OS X versions is still in the hands of the vendor.
According to this page your printer is not only still supported but the software has been updated recently. Not sure where your info
it doesn't come any longer with Apple updates
is from. Have you run Software Update recently? If you did and the printer drivers did not get updated, you'll get the latest HP drivers here.