Dear bauch,

Good a.m. and may this find you well. Look at the On-Board Diagnostics Codes.
Things to check:
(a) The Idle Air Control Valve (mounted on a bracket, next to the intake manifold) may need cleaning. When it gets dirty, the valve sticks, producing rough idle. Cold may simply cause oil deposits to become "gummy"; they don't do so at higher temps. See the FAQs for the cleaning procedure.
(b) Failing ECU temperature sensor. This is the rear-most of two, mounted in the head, on the same side as the intake manifold. Hard to access. Its signal tells the ECU, when the engine has warmed to normal operating temperature. If it fails, there should be some OBD codes
(c) It is possible that the RPM/Crank sensor is on the way out. This sensor, mounted atop the transmission bell housing - 10 o'clock position as you face the engine - is subject to failure, when the wiring harness insulation cracks, letting water in or allowing the wires to touch. Signal loss normally stops the engine totally, but brief contact could produce rough running. If this sensor has not been changed, it doesn't owe you a nickel. See the FAQs for the change procedure.

Hope this helps.

Yours faithfully,

Spook