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Lee and Pandharipande studied a "double point" algebraic cobordism theory ofvarieties equipped with vector bundles, and speculated that some features ofthat story might extend to the case of varieties with principal G-bundles. Thisnote shows that this expectation holds rationally, and more generally afterinverting the torsion index of the group, for reductive G. We show that (afterinverting the torsion index) the full theory for bundles on varieties is anextension of scalars of standard algebraic cobordism, that the theory for apoint is dual to Omega^*(BG), and describe how the theory for G compares tothat for a maximal torus T.

Algebraic cobordism of varieties with G-bundles