So I figured it out, I post it here in case somebody has the same issue.
You need to make a custom postprocess material. Just make a new material and do the same as you see on the attached screenshot.
Change the material domain to Post process, then scroll down to Post process material and change it to Before tonemapping. If you don’t do this, your new AO won’t get any AA and will look jaggy. Save it and create a Post process volume. In the Misc section look for the blendables, hit the plus sign and add your new material (as on the screenshot). Enjoy your new AO, don’t forget to tweak it in the PPVolume’s AO section. 