We investigate a mutual relationship between information and energy during the early phase of LTP induction and maintenance in a large-scale system of mutually coupled dendritic spines, with discrete internal states and probabilistic dynamics, within the framework of nonequilibrium stochastic thermodynamics. In order to analyze this computationally intractable stochastic multidimensional system, we introduce a pair approximation, which allows us to reduce the spine dynamics into a lower-dimensional manageable system of closed equations. We found that the rates of information gain and energy attain their maximal values during an initial period of LTP (i.e., during stimulation), and after that, they recover to their baseline low values, as opposed to a memory trace that lasts much longer. This suggests that the learning phase is much more energy demanding than the memory phase. We show that positive correlations between neighboring spines increase both a duration of memory trace and energy cost during LTP, but the memory time per invested energy increases dramatically for very strong, positive synaptic cooperativity, suggesting a beneficial role of synaptic clustering on memory duration. In contrast, information gain after LTP is the largest for negative correlations, and energy efficiency of that information generally declines with increasing synaptic cooperativity. We also find that dendritic spines can use sparse representations for encoding long-term information, as both energetic and structural efficiencies of retained information and its lifetime exhibit maxima for low fractions of stimulated synapses during LTP. Moreover, we find that such efficiencies drop significantly with increasing the number of spines. In general, our stochastic thermodynamics approach provides a unifying framework for studying, from first principles, information encoding, and its energy cost during learning and memory in stochastic systems of interacting synapses.