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Correction n+1 -> p+1 in the descent spectral sequence
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ACL
ACL
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As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1], $$[X/G]_n = (G^{n+1}\times X)/G$$ One has a descent spectral sequence $$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{n+1}\times H^q(X,\mathbb{Q}))/G$$$$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{p+1}\times H^q(X,\mathbb{Q}))/G$$ abutting to $H^{p+q}([X/G]_\bullet, \mathbb{Q})=H^{p+q}_G(X,\mathbb{Q})$ [Hodge III, (5.2.1.1)], and this is compatible with MHS [Hodge III, (8.1.15)]. Now use the fact that the complex $E_1$ is the bar complex, which computes group cohomology; in this case it is trivial except in degree zero. So in conclusion $$H_G^*(X,\mathbb{Q})\cong H^*(X,\mathbb{Q})^G$$ as MHS.

As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1], $$[X/G]_n = (G^{n+1}\times X)/G$$ One has a descent spectral sequence $$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{n+1}\times H^q(X,\mathbb{Q}))/G$$ abutting to $H^{p+q}([X/G]_\bullet, \mathbb{Q})=H^{p+q}_G(X,\mathbb{Q})$ [Hodge III, (5.2.1.1)], and this is compatible with MHS [Hodge III, (8.1.15)]. Now use the fact that the complex $E_1$ is the bar complex, which computes group cohomology; in this case it is trivial except in degree zero. So in conclusion $$H_G^*(X,\mathbb{Q})\cong H^*(X,\mathbb{Q})^G$$ as MHS.

As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1], $$[X/G]_n = (G^{n+1}\times X)/G$$ One has a descent spectral sequence $$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{p+1}\times H^q(X,\mathbb{Q}))/G$$ abutting to $H^{p+q}([X/G]_\bullet, \mathbb{Q})=H^{p+q}_G(X,\mathbb{Q})$ [Hodge III, (5.2.1.1)], and this is compatible with MHS [Hodge III, (8.1.15)]. Now use the fact that the complex $E_1$ is the bar complex, which computes group cohomology; in this case it is trivial except in degree zero. So in conclusion $$H_G^*(X,\mathbb{Q})\cong H^*(X,\mathbb{Q})^G$$ as MHS.

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Donu Arapura
Donu Arapura
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As I said in my comment, the mixed Hodge structures are the same. Here is the outline. From [Hodge III, 6.1.2.1], $$[X/G]_n = (G^{n+1}\times X)/G$$ One has a descent spectral sequence $$E_1= H^q([X/G]_p,\mathbb{Q})\cong (G^{n+1}\times H^q(X,\mathbb{Q}))/G$$ abutting to $H^{p+q}([X/G]_\bullet, \mathbb{Q})=H^{p+q}_G(X,\mathbb{Q})$ [Hodge III, (5.2.1.1)], and this is compatible with MHS [Hodge III, (8.1.15)]. Now use the fact that the complex $E_1$ is the bar complex, which computes group cohomology; in this case it is trivial except in degree zero. So in conclusion $$H_G^*(X,\mathbb{Q})\cong H^*(X,\mathbb{Q})^G$$ as MHS.