In thermodynamics, extensity x is shorthand term for an "extensive quantity", a quantity proportional to a system’s dimensions, having the conjugate variable tension X.  Internal energy U is the thermodynamic potential for systems when quantities of extensity are constant.

Tension
A tension is a partial derivative of the internal energy U with respect to an extensity, other extensive quantities being kept constant. Subsequently, with any extensity xi it is always possible to associate a tension variable Xi: $X_i = \frac{\partial U}{\partial x_i}$

which is called the conjugate. As such, according to the first law, the change in internal energy of a system is given by: $dU = \sum_{i=1}^k X_i dx_i$

Stated verbially, extensity is an energy “transfer variable”, defined such that when two systems A and B are brought into contact or interact, the assembly (A + B) being isolated, with each system having different tensions, an exchange of energy results via a transfer of an extensity, such that the transfer ends when the tensions have equalized.  Extensity is conserved for reversible transformation. 

Psychological thermodynamics
See main: Psychological thermodynamics
At some point, the term extensity of thermodynamics seems to have crossed over into the field of psychodynamics, possibly through the writings of Swiss psychiatrist Carl Jung or a terminology borrowed from Belgian chemist Ilya Prigogine. In this sense, extensity, according to one definition, is an ordering parameter of a complex system, associated with energy, defined as the area of consciousness in phase space; where phase space for the ego is a two-dimensional chart showing the relationships between consciousness, the personal unconsciousness, and the collective unconsciousness over time. 

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics. Oxford: Oxford University Press.
2. Richet, Pascal. (2001). The Physical Basis of Thermodynamics: with Applications to Chemistry, (pg. 11). Springer.
3. Schueler, Gerald J. and Schueler, Betty J. (2006). The Chaos of Jung’s Psyche, (Glossary) (T.O.C.). Online book: Schuelers.com.