'Just' the single glazing may not account for the cold you feel from there.
First, I would check and replace the weather stripping/draught excluder tape to a thickness that really squishes when you close the door. Check the threshold too.
Second, I would look at if the putty/beading is leaking air and needs redoing. I recently replaced the putty on one of my doors (with butyl tape) and the whole thing feels a lot more substantial. While you're there, you could see if it could take a thicker pane of glass.
Third, check the joins of the door. A few years ago at my last rental property, I filled/glued some gapping joints and repainted a door, and it totally changed the feel from flimsy to solid. The weather stripping somehow seemed to work better too.
Fourth, there's room attach a rail above the door frame to hang a nicely heavyweight curtain there.
Wood can be a really good insulator and the door looks in decent enough condition, so unless the owner is flush, they may not want to replace it.