Could thermal fluctuations seed cosmic structure?

We examine the possibility that thermal, rather than quantum, fluctuations are responsible for seeding the structure of our universe. We find that while the thermalization condition leads to nearly Gaussian statistics, a Harrisson-Zeldovich spectrum for the primordial fluctuations can only be achieved in very special circumstances. These depend on whether the universe gets hotter or colder in time, while the modes are leaving the horizon. In the latter case we find a no-go theorem which can only be avoided if the fundamental degrees of freedom are not particlelike, such as in string gases near the Hagedorn phase transition. The former case is less forbidding, and we suggest two potentially successful ``warming universe'' scenarios. One makes use of the Phoenix universe, the other of ``phantom'' matter.