A319 Captain reported cargo door amber light resulted in return to departure airport.

Narrative

After Takeoff as we were climbing we got a DOOR (AFT) CARGO amber light at around 3;000 to 4;000ft. The FO took over the radios and CA ran the after take off checklist. Immediately ran the ECAM for the DOOR AFT CARGO which advised to Limit maximum flight level to FL 10;000ft. We were cleared to 12;000ft but told ATC that we had to remain no higher than 10;000ft. We first climbed up to about 8;000 to 7;000 feet. The CA was on the ACARs with Dispatch informing that we would land back in ZZZ since we could not continue due to this ECAM. As we were climbing between 6;000 and 7;000ft the FO noticed that the aircraft was not pressurizing at all so we descended to a lower altitude. After telling the dispatch and ATC of intentions to return to ZZZ the CA advised ATC on the radio. We then started to prepare to fly back to ZZZ. The CA informed the FA and PAX (passengers) about the diversion. Then FMC was setup up to land at XXR in ZZZ and we ran the appropriate decent and landing checklist. We decided to call ARFF (Airport Rescue and Firefighting) after landing so that they can look at the Aft Cargo door to make sure it was closed and secured. Inspection revealed that the door was closed. We then Taxied to the gate. We also ran the landing overweight checklist prior to landing and let MX (Maintenance) know that it was an overweight landing.