Say we had a circuit with a cell (with internal resistance)and an external resistor of $10 \Omega$ as shown:
The $10\Omega$ resistor then increases in temperature, leading to an increase in resistance in this resistor. My thinking was that since $10 \Omega$ resistor now has a value higher than $10\Omega$, more voltage drops across this resistor. Since your emf is constant, less voltage must drop across the internal resistor. Less voltage drop means less resistance. Where is the flaw in my logic?