We introduce double Bott-Samelson cells defined by a symmetrizable
generalized Cartan matrix $C$ and a pair of positive braids $(b,d)$ in the
braid group associated to $C$. We prove that the coordinate rings of double
Bott-Samelson cells are upper cluster algebras, and prove the Fock-Goncharov
cluster duality conjecture in these cases. We construct the Donaldson-Thomas
transformation on double Bott-Samelson cells and prove that it is a cluster
transformation. We prove a periodicity phenomenon of the cluster
Donaldson-Thomas transformation on a family of double Bott-Samelson cells
associated to Cartan matrices of finite type. As an application, we obtain a
new geometric proof of Zamolodchikov's periodicity conjecture in the cases of
$\Delta\otimes \mathrm{A}_n$.