The rebound effect is a counter response to the use of a good after its use becomes more efficient. For example after investing in energy efficiency or the introduction of a more efficient technology.
After investing in energy efficiency the cost of energy drops that much, that it again becomes attractive to use more of it. A new technology that increases efficiency, and thus lowers costs, could also cause the rebound effect. In that case the extra remaining income will be partially spend on energy use again.
Users of energy have, mostly unconsciously, a budget for energy use, which means the mathematically expected drop in costs will not be achieved. A drop in the price per unit raises demand for most goods and services. When through higher efficiency the use of something becomes cheaper, you expect the rebound effect to apply. People will raise the temperature from the heater because energy use became less costly.
Do the investments that were made play a role? In most cases only when the decision to invest is made. In many cases it is hard to fix prior behavior, in other words leave the heater at a lower temperature. Once the investment is made, that could be considered a sunk cost, which in principle does not influence subsequent decisions. That is another reason why the rebound effect is expected when people invest in energy efficiency.