We provide a description of the Marrucci-Ianniruberto constitutive equation [Philos. Trans. R Soc. London, A 2003, 361, 677-688] for the rheology of entangled polymer melts in the context of nonequilibrium thermodynamics and we properly extend it to account for a second normal stress difference by introducing a second order term in the relaxation tensor in terms of the conformation tensor. The modified model incorporates one additional parameter, the anisotropic mobility parameter a, which allows for nonvanishing predictions of the second normal stress coefficient. Application of the second law of thermodynamics and the requirement that the evolution equation must preserve the positive-definite nature of the conformation tensor between successive entanglement points along the chain for all times and all flow fields constrain the convective constraint release (CCR) parameter beta(ccr) to values strictly greater than one (beta(ccr) > 1) and the new parameter alpha to values in the interval 0 <= alpha <= G 1- beta(-1)(ccr). The modified model provides a satisfactory description of available experimental data for the transient and steady-state shear rheology of entangled polystyrene melts [Schweizer et al. J. Rheol. 2004, 48, 1345-1363] and for the elongational steady-state stress of an entangled polystyrene solution [Ye et al. J. Rheol. 2003, 47, 443-468] over the entire range of shear and elongation rates covered in the rheological measurements.