Thermodynamics of a short-range model of spin ice magnets in a field is considered in the Bethe-Peierls approximation. The results obtained for [111], [100], and [011] fields agree reasonably well with the existing Monte Carlo simulations and some experiments. In this approximation, all extremely sharp field-induced anomalies are described by analytic functions of temperature and the applied field. In spite of the absence of true phase transitions, the analysis of the entropy and specific heat reliefs over the H-T plane allows discerning "pseudo-phases" with a specific character of spin fluctuations and defining the lines of relatively sharp "pseudo- transitions" between them.
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