Increasing the projection order in the affine projection adaptive filtering algorithm speeds up the convergence but also increases the steady-state misalignment. To address this unfavorable compromise, we propose a new affine projection algorithm with a variable projection order. This algorithm adaptively changes the projection order according to the estimated variance of the filter output error. The error variance is estimated using the exponential window and moving averaging techniques and employing a variable forgetting factor. Simulations demonstrate that the new algorithm provides fast initial convergence and low steady-state misalignment without necessarily trading off one for the other in addition to a significant reduction in average computational complexity.