Yes, the annealing temperature is important for the binding of the primers on the DNA. If you increase this temperature, you will decrease the binding of the primers and decrease the Ct value.
If the amplification of the cDNA for the same primer/gene did not vary with the increase in annealing temperature then the ct value never varies. The ct value is purely depends on the amplification so the amplification should be perfect with the given temperature. If you have any doubts with the ct value check the melt curve.
You should try a gradient of annealing temperature to find the best one, than your ct value automatically will be ok. To do this you need a PCR machine that allows you to play with Tm gradients.
It is just a matter of setting up your machine. The one we have is a CFX96 from bio-rad. http://www.bio-rad.com/pt-br/category/real-time-pcr-detection-systems.