  This present paper is dedicated to the study of the Cauchy problem of the
two-dimensional Euler-Boussinesq-B$\rm\acute{e}$nard equations which couple the
incompressible Euler equations for the velocity and a transport equation with
critical dissipation for the temperature. We show that there is a global unique
solution to this model with Yudovich's type data. This settles the global
regularity problem which was remarked by Wu and Xue (J. Differential Equations
253:100--125, 2012).
