The JTS rootservices document shows this:

  <lpa:UserProjects rdf:resource="https://localhost:9443/jts/lifecycle-project/user-projects"/>
    

A GET on that URI returns all the LPA projects. For example:

<rdf:RDF
    xmlns:lpa="http://jazz.net/xmlns/foundation/lpa/1.0/"
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#">
  <lpa:Project rdf:about="https://localhost:9443/jts/lifecycle-project/projects/_QdsbQWzMEe256eutwyd3fg">
    <dcterms:title>Money that Matters Sample</dcterms:title>
    <lpa:container rdf:resource="https://localhost:9443/rm/process/project-areas/_L3574GzMEe2Ygu8NrMBXLQ"/>
    <lpa:container rdf:resource="https://localhost:9443/qm/process/project-areas/_NgS2J2zMEe2qf8MTbnQFLQ"/>
    <lpa:container rdf:resource="https://localhost:9443/ccm/process/project-areas/_MgbUhGzMEe2haqWJ7abTEw"/>
  </lpa:Project>
  <lpa:Project rdf:about="https://localhost:9443/jts/lifecycle-project/projects/_PDd8K2zQEe256eutwyd3fg">
    <dcterms:title>Systems Global Configuration Sample</dcterms:title>
    <lpa:container rdf:resource="https://localhost:9443/rm/process/project-areas/_MXOYUWzQEe2Ygu8NrMBXLQ"/>
    <lpa:container rdf:resource="https://localhost:9443/qm/process/project-areas/_NCaDUWzQEe2qf8MTbnQFLQ"/>
    <lpa:container rdf:resource="https://localhost:9443/gc/process/project-areas/_N7BGUGzQEe2I-Orn6O9qBw"/>
    <lpa:container rdf:resource="https://localhost:9443/ccm/process/project-areas/_MoHVjGzQEe2haqWJ7abTEw"/>
  </lpa:Project>
  <lpa:Container rdf:about="https://localhost:9443/rm/process/project-areas/_MXOYUWzQEe2Ygu8NrMBXLQ">
    <dcterms:title>JKE Banking SGC (Requirements Management)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/gc/process/project-areas/_N7BGUGzQEe2I-Orn6O9qBw">
    <dcterms:title>JKE Banking SGC (Global Configuration)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/qm/process/project-areas/_NCaDUWzQEe2qf8MTbnQFLQ">
    <dcterms:title>JKE Banking SGC (Quality Management)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/ccm/process/project-areas/_MoHVjGzQEe2haqWJ7abTEw">
    <dcterms:title>JKE Banking SGC (Change Management)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/ccm/process/project-areas/_MgbUhGzMEe2haqWJ7abTEw">
    <dcterms:title>JKE Banking (Change Management)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/rm/process/project-areas/_L3574GzMEe2Ygu8NrMBXLQ">
    <dcterms:title>JKE Banking (Requirements Management)</dcterms:title>
  </lpa:Container>
  <lpa:Container rdf:about="https://localhost:9443/qm/process/project-areas/_NgS2J2zMEe2qf8MTbnQFLQ">
    <dcterms:title>JKE Banking (Quality Management)</dcterms:title>
  </lpa:Container>
</rdf:RDF>
    

As you can see, it includes very little meta data about the projects.

However, it appears you can do a GET on a specific project. For example, a GET on https://localhost:9443/jts/lifecycle-project/projects/_QdsbQWzMEe256eutwyd3fg returned more detailed information (in JSON only it seems), that did seem to include a modified data for each contributed project area. I didn't see offhand any modified property for the whole LPA project.

I couldn't find any documentation for this referenced from https://jazz.net/wiki/bin/view/Deployment/CLMProductAPILanding, so it's possible that all of these are private undocumented APIs. In which case, use at your own risk.