  We prove strong rate resp. weak rate ${\mathcal O}(\tau)$ for a structure
preserving temporal discretization (with $\tau$ the step size) of the
stochastic Allen-Cahn equation with additive resp. multiplicative colored noise
in $d=1,2,3$ dimensions. Direct variational arguments exploit the one-sided
Lipschitz property of the cubic nonlinearity in the first setting to settle
first order strong rate. It is the same property which allows for uniform
bounds for the derivatives of the solution of the related Kolmogorov equation,
and then leads to weak rate ${\mathcal O}(\tau)$ in the presence of
multiplicative noise. Hence, we obtain twice the rate of convergence known for
the strong error in the presence of multiplicative noise.
