Was the PETM Linked to Hydrothermal Venting? 

The PETM (Palaeocene-Eocene Thermal Maximum) was a period of rapid global warming. Sea surface temperatures rose 6°C within 10 to 20 thousand years. This rate of temperature rise is not very different to what is happening currently. As THIS ARTICLE says, the temperature rise led to the release of organic carbon into the atmosphere. Expressed as CO₂, 0.24 gigatons (Gt) per year. Currently we release 0.51Gt per year. The article is based on THIS PDF.

In the PETM the temperature rise led to the most dramatic biologic changes since the asteroid killed the dinosaurs.

What was the cause? At the time there was magmatic activity around the area which was to become the Atlantic. This was the start of continental rifting. Much of the activity was the intrusion of sills. Could the sills have led to the release of methane from organic rich sediments?

The article suggests that is what happened. Using a 3-D seismic survey they found pockmarks beneath the P-E boundary. Detailed study suggests they formed from hydrothermal venting. 

Because they happened in shallow water the CH₄ went straight into the atmosphere. (At depth it would have been mitigated by the methane being oxidised to carbon dioxide.)

Three-dimensional view of seismic reflection data off western Norway. The greytone lower part is a vertical ‘slice’. The coloured part shows the depth variation of sediments that fill hydrothermal vent systems beneath a horizontal unconformity. (Credit: Berndt et al, Fig 1b)