  Several rare earth magnetic pyrochlore materials are well modeled by a
spin-1/2 quantum Hamiltonian with anisotropic exchange parameters Js. For the
Er2Ti2O7 material, the Js were recently determined from high-field inelastic
neutron scattering measurements. Here, we perform high-temperature (T) series
expansions to compute the thermodynamic properties of this material using these
Js. Comparison with experimental data show that the model describes the
material very well including the finite temperature phase transition to an
ordered phase at Tc~1.2 K. We show that high temperature expansions give
identical results for different q=0 xy order parameter susceptibilities up to
8th order in \beta=1/T (presumably to all orders in \beta). Conversely, a
non-linear susceptibility related to the 6th power of the order parameter
reveals a thermal order-by-disorder selection of the same non-colinear \psi_2
state as found in Er2Ti2O7.
