  We develop the finite temperature theory of p-adic string models. We find
that the thermal properties of these non-local field theories can be
interpreted either as contributions of standard thermal modes with energies
proportional to the temperature, or inverse thermal modes with energies
proportional to the inverse of the temperature, leading to a "thermal duality"
at leading order (genus one) analogous to the well known T-duality of string
theory. The p-adic strings also recover the asymptotic limits (high and low
temperature) for arbitrary genus that purely stringy calculations have yielded.
We also discuss our findings surrounding the nature of the Hagedorn transition.
