  Two dimensional quantum R$^2$-gravity is formulated in the semiclassical
method. The thermodynamic properties,such as the equation of state, the
temperature and the entropy, are explained. The topology constraint and the
area constraint are properly taken into account. A total derivative term and an
infrared regularization play important roles. The classical solutions (vacua)
of R$^2$-Liouville equation are obtained by making use of the well-known
solution of the ordinary Liouville equation. The positive and negative constant
curvature solutions are 'dual' each other. Each solution has two
branches($\pm$). We characterize all phases. The topology of a sphere is mainly
considered.
