In this paper, we address the challenge of adaptively controlling a home heating system in order to minimise cost and carbon emissions within a smart grid. Our home energy management agent learns the thermal properties of the home, and uses Gaussian processes to predict the environmental parameters over the next 24 hours, allowing it to provide real time feedback to householders concerning the cost and carbon emissions of their heating preferences. Furthermore, we show how it can then use a mixed-integer quadratic program, or a computationally efficient greedy heuristic, to adapt to real-time cost and carbon intensity signals, adjusting the timing of heater use in order to satisfy preferences for comfort whilst minimising cost and carbon emissions. We evaluate our approach using weather and electricity grid data from January 2010 for the UK, and show our approach can predict the total cost and carbon emissions over a day to within 9%, and show that over the month it reduces cost and carbon emissions by 15%, and 9%, respectively, compared to using a conventional thermostat.