If some diode laser is given an input current, why should we see mode-hopping if the current is changed?
I'm not 100% sure, but I think that temperature is usually the primary culprit. More current means more resistive heating.
You can't just use a temperature controller to hold the temperature constant: The laser doesn't have just one temperature -- it has hotter parts and colder parts. (The resistive heating only occurs in certain areas within the laser.) No matter how much money you spend on a temperature controller, you cannot hold the laser's entire temperature profile constant when you change the laser current.
When the temperature profile changes, the refractive index profile changes too, therefore the modes' shape and frequency change, which changes the gains and losses of each mode in a different way. Therefore, mode hopping can occur.
EM waves can have many different modes of propagation and when a single mode laser is driven at a different current level sometimes these conditions are met allowing other modes of oscillation.
In multi-mode operation of a laser, there may be transitions between different sets of modes. However, simultaneous oscillation on many modes is then most common, and instead of complete mode hops there are often more continuous transitions, with the optical power being gradually redistributed. The dynamics can be further influenced by nonlinear effects such as spatial hole burning. Mode hops can also involve higher-order modes, or modes with different polarization in lasers with polarization-independent gain.
Typically when a diode is driven at a different power level the previously lasing mode may no longer be the mode with highest gain, so that the power of a competing mode with higher gain can quickly rise causing modes to form or switch. Also a drift of the temperature of the gain medium will shift the wavelength of maximum gain while not shifting the frequencies of the resonator modes.