That is true, but it depends on the individual tire types and what you consider 'cold'. Performance tires are designed to run in an optimum temperature range, typically well over 100degrees F (usually 120-140F, in my experience). Anything below that, and the rubber won't be soft enough to fully settle into the contours of the pavement. Anything above optimum, and the rubber shears away before maximum grip is reached, making the tires feel 'greasy'.
Most of the extreme performance summer tires have warnings that they are not designed to be used in freezing temperatures. If by "competition tires", you mean "D.O.T. R-compund tires (Hoosier A6/R6, Kumho 710, etc), exposing them to below freezing temperatures damages them by hardening the rubber.
If it helps; most of the nationally competitive AutoXers in street tire categories have two or more sets of tires around and let the weather decide for them. Hankook RS-3s seem to be the favorite for warm weather (About 80F ambient, and above), and Toyo R1Rs for colder weather and rain. It is always funny when the temps are around 80 degrees and drivers pull into grid with Hankooks and Toyos. Toyo drivers will be spraying their tires with water, while the Hankook drivers next to them are covering their tires to retain the heat.