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The Bott-Thurston cocycle is a $2$-cocycle on the group of
orientation-preserving diffeomorphisms of the circle. We introduce and study a
formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a
$2$-cocycle on the group of continuous $A$-automorphisms of the algebra
$A((t))$ of Laurent series over a commutative ring $A$ with values in the group
$A^*$ of invertible elements of $A$. We prove that the central extension given
by the formal Bott-Thurston cocycle is equivalent to the $12$-fold Baer sum of
the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a
consequence of this result we prove a part of new formal Riemann-Roch theorem.
This Riemann-Roch theorem is applied to a ringed space on a separated scheme
$S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally
on $S$ isomorphic to the sheaf ${\mathcal O}_S((t))$ and the transition
automorphisms are continuous. Locally on $S$ this ringed space corresponds to
the punctured formal neighbourhood of a section of a smooth morphism to $U$ of
relative dimension $1$, where an open subset $U \subset S$.