Perhaps is just a modelling problem. For example, in this paper [http://arxiv.org/pdf/1407.6191v1.pdf] the author calculate the work done by a noisy correlated external force. My assumption is that because this force is not purely white they related it to a 'oriented' injection of energy in the system. But for me it is not clear why this external force shouldn't contribute to his heat definition Eq. (4) as it introduces randomness. My conclusion is that his definition is changed in the limit that his external force have vanishing correlation.

Thank you for your answer. 

The notion of temperature is problematic if the system is not in thermodynamic equilibrium. At thermodynamic equilibrium different sources of noise are normally uncorrelated. Therefore your question may not have any proper answer. At least you need have to specify your system in more detail if there should be a meaningful answer. For instance your proposal might be valid if the correlation is very weak so that the system is close to equilibrium.

Dear Gert,

your calculation seems to state well my concern. This is very nice, thank you for taking your time. But, right now I'm taking a look at the experiment of Ciliberto. Note that the noise bath in the supplementary material have a colored spectrum.