All coronaviruses possess a fat layer as a component of the viral envelope. There is, however, no evidence yet that SARS-CoV-2 (the virus that causes COVID-19) exhibits structural stability at low temperatures or that its fat envelope facilitates transmission at low temperatures.
This claim was likely derived from studies showing that SARS-CoV (the SARS virus from the early 2000s, not the one that causes COVID-19) exhibits structural stability at relatively low temperatures (22–25°C), though it was never shown that this was associated with it being "transmitted [between people or between surfaces and people] better in the cold."
In fact, enveloped viruses with a fat membrane are usually more sensitive to neutralizing agents (like chemical antiseptics) and conditions (like changes in temperature or pH) than non-enveloped viruses.
Drafted 23 March 2020