  A time-fractional Allen-Cahn equation with volume constraint is first
proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive
linear second-order energy stable schemes are developed for the proposed model
by combining invariant energy quadratization and scalar auxiliary variable
approaches with the recent L1$^{+}$ formula. The new developed methods are
proved to be volume-preserving and unconditionally energy stable on arbitrary
nonuniform time meshes. The accelerated algorithm and adaptive time strategy
are employed in numerical implement. Numerical results show that the proposed
algorithms are computationally efficient in multi-scale simulations, and
appropriate for accurately resolving the intrinsically initial singularity of
solution and for efficiently capturing the fast dynamics away initial time.
