Topology optimization is an optimization method which can modify connectivity of an object independently of its predefined topology. In this paper, a global optimization method for topology optimization of flow channels considering fluid and heat transfer using a genetic algorithm is presented. A genetic algorithm (GA) is assisted by the Kriging surrogate model to reduce computational cost required for function evaluation. In the present method, the boundary of a flow channel is represented by a level set function. Topology optimization seldom employs GA since topology optimization requires a large number of design variables for a high degree of freedom for shape and topology representation and GA is not effective to handle such a large scale problem. This paper presents a novel representation method to obtain the distribution of level set function with a reasonable number of design variables. The design variables are given at the scattered control points in the design domain, and the Helmholtz equation is solved in the entire domain. The proposed method is applied to a single-objective optimization problem to maximize heat transfer. As a result, GA found several flow channels, each of which has similar objective function values but with different topology. The result indicates that the objective function is a multi-modal function, which means that a method of population-based multipoint simultaneous exploration such as GA is essential for the present topology optimization problem. Considering minimizing pressure loss of a flow channel as the second objective function, the proposed method is applied to a multi-objective optimization problem. As a result, we confirm that the proposed representation method enables to represent flow channels that balance both objective functions and GA captures the trade-off between two objective functions.