  We discuss a $\lambda\varphi^{4}+\rho\varphi^{6}$ scalar field model defined
in the Euclidean section of the Schwarzschild solution of the Einstein
equations in the presence of multiplicative noise. The multiplicative random
noise is a model for fluctuations of the Hawking temperature. We adopt the
standard procedure of averaging the noise dependent generating functional of
connected correlation functions of the model. The dominant contribution to this
quantity is represented by a series of the moments of the generating functional
of correlation functions of the system. Positive and negative effective
coupling constants appear in these integer moments. Fluctuations in the Hawking
temperature are able to generate first-order phase transitions. Using the
Gaussian approximation, we compute $\langle\varphi^{2}\rangle$ for arbitrary
values of the strength of the noise. Due to the presence of the multiplicative
noise, we show that $\langle\varphi^{2}\rangle$ near the horizon must be
written as a series of the the renormalized two-point correlation functions
associated to a free scalar field in Euclidean Rindler manifold.
