We study the canonical heat flow (Ht)t≥0 on the cotangent module L2(T⁎M) over an RCD(K,∞) space (M,d,m), K∈R. We show Hess–Schrader–Uhlenbrock's inequality and, if (M,d,m) is also an RCD⁎(K,N) space, N∈(1,∞), Bakry–Ledoux's inequality for (Ht)t≥0 w.r.t. the heat flow (Pt)t≥0 on L2(M). Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various Lp-properties of (Ht)t≥0, p∈[1,∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian Δ→, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ+K. In the RCD⁎(K,N) case, we prove compactness of Δ→−1 if M is compact, and the independence of the Lp-spectrum of Δ→ on p∈[1,∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (Ht)t≥0. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it.