  We discuss the non-constant dilaton deformed ${\rm AdS}_5\times{\rm S}_5$
solutions of IIB supergravity where AdS sector is described by black hole. The
investigation of running gauge coupling (exponent of dilaton) of non-SUSY gauge
theory at finite temperature is presented for different regimes (high or low T,
large radius expansion). Running gauge coupling shows power-like behavior on
temperature with stable fixed point. The quark-antiquark potential at finite T
is found and possibility of confinement is established. It is shown that
non-constant dilaton affects the potential, sometimes reversing its behavior if
we compare it with the constant dilaton case (${\cal N}=4$ super Yang-Mills
theory). Thermodynamics of obtained backgrounds is studied. In particular,
next-to-leading term to free energy F is evaluated as $F=-{\tilde V_3 \over
4\pi^2}({N^2 (\pi T)^4 \over 2} + {5c^2 \over 768 g_{YM}^6 N{\alpha'}^6 (\pi
T)^4})$. Here $\tilde V_3$ is the volume of the space part in the boundary of
AdS, c is the parameter coming from the non-constant dilaton and N is the
number of the coincident D3-branes.
