There is a definition of $h^{1}(\overline{D})$ by many people, and a definition of $H^{1}(\overline{D})$ by Alexrander Borisov using the notion of ghost spaces of the second kind. But neither is the same as the "original ones" you are talking about.

As Neukirch pointed out in his book (page 210), in the classical setting an analgous definition of $H^{1}(\overline{D})$ is completely missing. The classical way of getting around this issue is to define $\chi(D)$ instead, and to generalize either you sacrifice the exactness of Riemann-Roch or you re-write the theory using $K$- groups. Both have been extensively discussed in Neukirch. As far as I know, apparently the same situation happens for two dimensional Arakelov theory over surfaces as well: we do not have a reasonable definition of $H^{i}$, instead we have a "Faltings volume" that can be used in some ways to form the Euler characteristic. There has been works done by many people to generalize this to higher dimensional situations in 1980s using index theorem, including efforts replacing Faltings volume by Quillen metric and substantial use of characteristic classes.

If we step out of the comfort zone and willing to compromise, then there are "regularization procedure" that allows you to define $h^{1}(\overline{D})$ which essentially boils down to Poisson summation formula, or Pontrajin duality if you prefer. And it has been invented and re-invented by many people. But the essential obstacle is still the same. In particular it seems rather unlikely that we will have an Arakelov cohomology theory built upon derived functors or Cech cohomology alone in the case of arithemetic surface. But we do have something analgous to Serre-duality proved by Borisov in this case, and generalized to the case of vector bundles using a construction similar to adeles by Ichiro Miyada. However the higher dimensional analog is missing.

The difficulty seems to be concentrated on the particular case of the vertical divisor at infinity, for which the "ghost space approach" collapses and classical approach thrives (see Lang, page 114). In all other cases (horizontal divisor, vertical divisor at finite places) the ghost space gives what we wanted. In particular, it is not clear how to define a united theory that can recover Faltings-Riemann-Roch in two dimensional case. There are quite a few Arakelov theory experts in the forum (Minhyong Kim, for example) who wrote their theses around this topic. I am sure they can provide much more professional feedback.

Update:

I learned it from Professor Soule, but I think this is well known among experts in the research circle, that the Faltings' volume has an explicit relationship with Quillen's metric. The exact formula may be found from Professor Soule's Bourbaki lecture paper on arithemetic surfaces. If I am not mistaken there is no higher dimensional analog of Falting's volume for general arithemetic varieties. As far as I know, current work attempting further generalizing Gillet-Soule's work (in the case of manifold with singularity, for example) uses analytic torsion instead. So far even very simple technical questions like "Can one derive analytic torsion using Faltings volume directly without using Faltings-Riemann-Roch?" can be difficult to answer.