A definition originally proposed by H. S. Green is used to calculate the entropy of nonequilibrium steady states. This definition provides a well-defined coarse graining of the entropy. Although the dimension of the phase space accessible to nonequilibrium steady states is less than the ostensible dimension of that space, the Green entropy is computed from within the accessible phase space, thereby avoiding the divergences inherent in the fine-grained entropy. It is shown that the Green entropy is a maximum at equilibrium and that away from equilibrium, the thermodynamic temperature computed from the Green entropy is different from the kinetic temperature.