We propose an orbital magnetothermal effect wherein a temperature gradient generates an orbital magnetization (OM) for Bloch electrons, and we present a unified theory for electrically and thermally induced OM, valid for both metals and insulators. We reveal that there exists an intrinsic response of OM, for which the susceptibilities are completely determined by the band geometric quantities such as interband Berry connections, interband orbital moments, and the quantum metric. The theory can be readily combined with first-principles calculations to study real materials. As an example, we calculate the OM response in CrI$_{3}$ bilayers, where the intrinsic contribution dominates. The temperature scaling of intrinsic and extrinsic responses, the effect of phonon drag, and the phonon angular momentum contribution to OM are discussed.
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