We performed a systematic study on the recoverable enthalpy in several glass-forming polymers. We found that after prolonged isothermal physical aging the enthalpy reaches a plateau with values substantially larger than than those corresponding to the enthalpy state extrapolated from the melt state. Enthalpy recovery experiments after up-jumps indicate that the enthalpy state corresponding to the plateau found after simple down-jump experiments is restored after long-term aging. This result is interpreted considering the plateau in the enthalpy as a thermodynamically stable state. We argue on the possible scenarios emerging from this conclusion. In particular, we discuss whether polymer glasses in the achieved thermodynamic state are stable over any time scale, or rather this corresponds to a relative minimum with further evolution at much larger time scales. Finally, the shift factor obtained from aging time temperature superposition of enthalpy recovery data was found to considerably gel-Fulcher-Tammann equation, normally adequate to describe the segmental mobility above the glass transition temperature (T-g). The deviation of thermodynamics and dynamics from the behavior expected extrapolating the behavior from above T-g has been analyzed within the Adam-Gibbs framework, which actually relates the relaxation time and a thermodynamic magnitude, namely the configurational entropy. It has been found that, at least semiquantitatively for most of the investigated polymers, the connection between dynamics and thermodynamics holds also below T-g.