Can a first order reaction change to a second order reaction under specific temperatures?
It's theoretically possible, though I don't know if there exists specific examples.

The first thing to recognize is that reactions don't really "switch" in an on/off fashion (at least not based on temperature). If there's two physically possible reactions, they both occur under all conditions ... theoretically. The main reason you see one and not the other is due to the comparative rates of reaction - one reaction might be so slow that no appreciable amount of product forms in reasonable timeframes.

So while the rates of reactions change based on temperature, as Zhe mentions, the way that rate changes in most cases (the Arrhenius equation) means that the two reactions aren't going to swap positions based on those considerations alone.

However, kinetics is not the only knob we're playing with here, when we change temperatures. We're also messing with the thermodynamics. Specifically, temperature is a factor in the free energy equation

$$\ce{\Delta G = \Delta H - T \Delta S }$$

If the two reactions have different $\Delta H$ (enthalpy) and $\Delta S$ (entropy) values, then we can change the spontaneity of the reactions. At low temperatures, something with a very favorable enthalpy will be spontaneous, even with a disfavorable entropy. Conversely, at high temperatures, something with a very favorable entropy will be spontaneous, even with a disfavorable enthalpy.

So if you have a system which can undergo first-order and second order reactions (for example, a molecule which has ) and choose your reactant such such that the first order reaction has a more favorable enthalpy and a less favorable entropy than the second order reaction (or vice versa), there will be a temperature where the thermodynamically favored product will go from that of the first-order reaction to that of the second order reaction. Whether this would be enough to manifest itself as an appreciable difference in product depends on the particular numbers.

While we're on the discussion, there is another technique related to controlling product outcome, that exploits the balance between the two factors already discussed. That's using thermodynamic versus kinetic reaction control. In this setup, you have one reaction which provides a more stable product, but due to things like steric hindrance the reaction rate is lower.  The other product forms faster, but the product is less stable. If you set up reaction conditions such as short reaction times at high temperature where the speed of the reaction is the dominant factor, the faster-made "kinetically controlled" product dominates. However, if you use conditions such as long reaction times at low temperatures, where the stability of the products is the dominant factor, the more stable "thermodynamicly controlled" product dominates.