Tunneling atomic force microscopy (TUNA) was used at ambient conditions to measure the current-voltage ($I$-$V$) characteristics at clean surfaces of highly oriented graphite samples with Bernal and rhombohedral stacking orders. The characteristic curves measured on Bernal-stacked graphite surfaces can be understood with an ordinary self-consistent semiconductor modeling and quantum mechanical tunneling current derivations. We show that the absence of a voltage region without measurable current in the $I$-$V$ spectra is not a proof of the lack of an energy band gap. It can be induced by a surface band bending due to a finite contact potential between tip and sample surface. Taking this into account in the model, we succeed to obtain a quantitative agreement between simulated and measured tunnel spectra for band gaps $(12 ldots 37)$,meV, in agreement to those extracted from the exponential temperature decrease of the longitudinal resistance measured in graphite samples with Bernal stacking order. In contrast, the surface of relatively thick graphite samples with rhombohedral stacking reveals the existence of a maximum in the first derivative $dI/dV$, a behavior compatible with the existence of a flat band. The characteristics of this maximum are comparable to those obtained at low temperatures with similar techniques.
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