In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' $H_{dR}^1(Z,\mu)$ is defined (Remark 4.44), where $(Z,d,\mu)$ is a doubling metric measure space satisfying a weak Poincare inequality. This group is bi-Lipschitz invariant and depends only on the measure class of $\mu$.
Has this group been studied beyond Cheeger's remark? Is anything known about the relationship between Cheeger's $H_{dR}^1$ and other first cohomologies of $Z$?