  The one-loop effective action in QED at zero and finite temperature is
obtained by using the worldline approach. The Feynman rules for the
perturbative expansion of the action in the number of derivatives are derived.
The general structure of the temperature dependent part of the effective action
in an arbitrary external inhomogeneous magnetic field is established. The
two-derivative term in the effective action for spinor and scalar QED in a
static magnetic background at $T\neq 0$ is calculated.
