  We describe the deformation cohomology of a symplectic groupoid, and use it
to study deformations via Moser path methods, proving a symplectic groupoid
version of the Moser Theorem. Our construction uses the deformation
cohomologies of Lie groupoids and of multiplicative forms, and we prove that in
the symplectic case, deformation cohomology of both the underlying groupoid and
of the symplectic groupoid have de Rham models in terms of differential forms.
  We use the de Rham model, which is intimately connected to the
Bott-Shulman-Stasheff double complex, to compute deformation cohomology in
several examples. We compute it for proper symplectic groupoids using vanishing
results; alternatively, for groupoids satisfying homological 2-connectedness
conditions we compute it using a simple spectral sequence. Finally, without
making assumptions on the topology, we constructed a map relating
differentiable and deformation cohomology of the underlying Lie groupoid of a
symplectic groupoid, and related it to its Lie algebroid counterpart via van
Est maps.
