  We develop the thermodynamics of field theories characterized by non-local
propagators. We analyze the partition function and main thermodynamic
properties arising from perturbative thermal loops. We focus on the p-adic
models associated with the tachyon phenomenology in string theories. We
reproduce well known features of these theories, but also obtain many new
results. In particular, we explain how to maintain consistency of such
non-local theories by avoiding the appearance of ghosts at finite temperature.
As a consequence of this fact, the vacuum energy in p-adic theories becomes
positive. It is also hierarchically suppressed, and we explore the parameter
space where it is consistent with the observed value of the cosmological
constant.
