Check for https string in this documentation and this one.
You should show us in your question your current ModPagespeedMapOriginDomain && ModPagespeedDomain settings.
From what I understand from these lines:
The origin_specified_in_html can specify https but the origin_to_fetch_from can only specify http, e.g.
ModPagespeedMapOriginDomain http://localhost https://www.example.com
This directive lets the server accept https requests for www.example.com without requiring a SSL certificate to fetch resources - in fact, this is the only way mod_pagespeed can service https requests as currently it cannot use https to fetch resources. For example, given the above mapping, and assuming Apache is configured for https support, mod_pagespeed will fetch and optimize resources accessed using https://www.example.com, fetching the resources from http://localhost, which can be the same Apache process or a different server process.
And these ones:
mod_pagespeed offers limited support for sites that serve content through https. There are two mechanisms through which mod_pagespeed can be configured to serve https requests:
- Use ModPagespeedMapOriginDomain to map the https domain to an http domain.
- Use ModPagespeedLoadFromFile to map a locally available directory to the https domain.
The solution would be something like that (or the one with ModPagespeedLoadFromFile)
ModPagespeedMapOriginDomain http://localhost https://www.example.com
BUT, the real problem for you is that apache does not directly receive the HTTPS requests as the hardware load balancer handle it on his own. So the mod-pagespeed output filter does not even know it was requested for an SSL domain. And when it modify the HTML content, applying domain rewrite maybe, it cannot handle the https case.
So... one solution (untested) would be using another virtualhost on the apache server, still HTTP if you want, dedicated to https handling. All https related urls (/login,/checkout,...) would then be redirected to this specific domain name by the hardware load balancer. Let's say http://secure.acb.com. This name is only in use between the load balancer and front apaches (and quite certainly apache should restrict access to this VH to the load balancer only).
Then in these http://secure.acb.com virtualhosts mod_pagespeed would be configured to externally rewrite domains to https://www.example.com. Something like:
ModPagespeedMapOriginDomain http://secure.example.com https://www.example.com
Finally the end user request is https://www.example.com/login, the load balancer manages HTTPS, talk to apache with http://secure.example.com, and page results contains only references to https://www.example.com/* assets. Now when theses assets are requested with an https domain request you still have the problem of serving theses assets. So the hardware load balancer should allow all theses assets url in the https domain and send them to the http://secure.abc.com virtualhosts (or any other static VH).
