Results from searches of double K capture in 78Kr in an experiment with the large-volume copper proportional counter, using data samples corresponding of two independent series of measurements with different intrinsic radioactivity background are presented. The total exposure of the low-background measurements is 0.343 kg×y. A combination of methods of selection of useful events with a unique set of characteristics and wavelet analysis of events allowed a reduction of the background by ∼2000 times in the energy region of interest. The statistical significance of combined data from two stages of operation equals 2.5σ. Corresponding to such effect, the half-life of 78Kr relative to 2ν2K capture equals T1/2=[9.2-2.6+5.5(stat)±1.3(syst)]×1021 y. Half-life limits for other 2K transitions to the excited states in 78Se are obtained at the level of 1021 y in the first time. In particular, limits on 2ν2K capture to the excited level 01+ (1499 keV) and resonant neutrinoless double K capture to the level 2+ (2838 keV) have been defined on the level of T1/2≥5.4×1021 y at 90% C.L.