The current 3.8σ tension between local [1] and global [2] measurements of H0 cannot be fully explained by the concordance ΛCDM model. It could be produced by unknown systematics or by physics beyond the standard model. In particular, non-standard dark energy models were shown to be able to alleviate this tension. On the other hand, it is well known that linear perturbation theory predicts a cosmic variance on the Hubble parameter H0, which leads to systematic errors on its local determination. Here, we study how including in the likelihood the cosmic variance on H0 affects statistical inference. In particular we consider the γCDM, wCDM and γwCDM parametric extensions of the standard model, which we constrain with the latest CMB, BAO, SNe Ia, RSD and H0 data. We learn two important lessons. First, the systematic error from cosmic variance is -independently of the model -approximately σcv ≈ 0.88 km s −1 Mpc −1 (1.2% H loc 0 ) when considering the redshift range 0.0233 ≤ z ≤ 0.15, which is relative to the main analysis of [1], and σcv ≈ 1.5 km s −1 Mpc −1 (2.1% H loc 0 ) when considering the wider redshift range 0.01 ≤ z ≤ 0.15. Although σcv affects the total error budget on local H0, it does not significantly alleviate the tension which remains at ≈ 3σ. Second, cosmic variance, besides shifting the constraints, can change the results of model selection: much of the statistical advantage of non-standard models is to alleviate the now-reduced tension. We conclude that, when constraining non-standard models it is important to include the cosmic variance on H0 if one wants to use the local determination of the Hubble constant by Riess et al. [1]. Doing the contrary could potentially bias the conclusions.