We introduce a stochastic growth model where the growth is controlled by a temperaturelike parameter T. The model shows various types of dynamical behavior as T changes from 0 to ∞. For T=0 the growth process belongs to the quenched Kardar-Parisi-Zhang (KPZ) universality class, whereas it belongs to the Edwards-Wilkinson (EW) universality class for T=∞. In the intermediate range 0<T<∞, the model shows an anomalous crossover behavior from the quenched KPZ to the thermal KPZ class. The KPZ nonlinearity is generated by an anisotropic effect of the quenched noise which exists only for T<∞ in our model. We also study crossovers between different types of scaling behavior of the interface width for various T’s.