  We explore the phase diagram of the 4D compact U(1) gauge theory at finite
temperature as a function of the gauge coupling and of the compactified
Euclidean time dimension L_t. We show that the strong-to-weak coupling
transition, which is first order at T=0 (L_t=\infty), becomes second order for
high temperatures, i.e. for small values of L_t, with a tricritical temporal
size \bar{L_t} located between 5 and 6. The critical behavior around the
tricritical point explains and reconciles previous contradictory evidences
found in the literature.
